Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oeeui Structured version   Visualization version   GIF version

Theorem oeeui 7569
 Description: The division algorithm for ordinal exponentiation. (This version of oeeu 7570 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 7552.) (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
oeeu.1 𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}
oeeu.2 𝑃 = (℩𝑤𝑦 ∈ On ∃𝑧 ∈ (𝐴𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
oeeu.3 𝑌 = (1st𝑃)
oeeu.4 𝑍 = (2nd𝑃)
Assertion
Ref Expression
oeeui ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜) ∧ 𝐸 ∈ (𝐴𝑜 𝐶)) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐶 = 𝑋𝐷 = 𝑌𝐸 = 𝑍)))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝑋,𝑦,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤)   𝐷(𝑥,𝑦,𝑧,𝑤)   𝑃(𝑥,𝑦,𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)   𝑋(𝑥)   𝑌(𝑥,𝑦,𝑧,𝑤)   𝑍(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem oeeui
Dummy variables 𝑎 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 3694 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
21adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐴 ∈ On)
32ad2antrr 758 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐴 ∈ On)
4 simprl 790 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐶 ∈ On)
5 oecl 7504 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
63, 4, 5syl2anc 691 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴𝑜 𝐶) ∈ On)
7 om1 7509 . . . . . . . . . . . . . . 15 ((𝐴𝑜 𝐶) ∈ On → ((𝐴𝑜 𝐶) ·𝑜 1𝑜) = (𝐴𝑜 𝐶))
86, 7syl 17 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴𝑜 𝐶) ·𝑜 1𝑜) = (𝐴𝑜 𝐶))
9 df1o2 7459 . . . . . . . . . . . . . . . 16 1𝑜 = {∅}
10 dif1o 7467 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (𝐴 ∖ 1𝑜) ↔ (𝐷𝐴𝐷 ≠ ∅))
1110simprbi 479 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ (𝐴 ∖ 1𝑜) → 𝐷 ≠ ∅)
1211ad2antll 761 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐷 ≠ ∅)
13 eldifi 3694 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ (𝐴 ∖ 1𝑜) → 𝐷𝐴)
1413ad2antll 761 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐷𝐴)
15 onelon 5665 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝐷𝐴) → 𝐷 ∈ On)
163, 14, 15syl2anc 691 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐷 ∈ On)
17 on0eln0 5697 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ On → (∅ ∈ 𝐷𝐷 ≠ ∅))
1816, 17syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (∅ ∈ 𝐷𝐷 ≠ ∅))
1912, 18mpbird 246 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ∅ ∈ 𝐷)
2019snssd 4281 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → {∅} ⊆ 𝐷)
219, 20syl5eqss 3612 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 1𝑜𝐷)
22 1on 7454 . . . . . . . . . . . . . . . . 17 1𝑜 ∈ On
2322a1i 11 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 1𝑜 ∈ On)
24 omwordi 7538 . . . . . . . . . . . . . . . 16 ((1𝑜 ∈ On ∧ 𝐷 ∈ On ∧ (𝐴𝑜 𝐶) ∈ On) → (1𝑜𝐷 → ((𝐴𝑜 𝐶) ·𝑜 1𝑜) ⊆ ((𝐴𝑜 𝐶) ·𝑜 𝐷)))
2523, 16, 6, 24syl3anc 1318 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (1𝑜𝐷 → ((𝐴𝑜 𝐶) ·𝑜 1𝑜) ⊆ ((𝐴𝑜 𝐶) ·𝑜 𝐷)))
2621, 25mpd 15 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴𝑜 𝐶) ·𝑜 1𝑜) ⊆ ((𝐴𝑜 𝐶) ·𝑜 𝐷))
278, 26eqsstr3d 3603 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴𝑜 𝐶) ⊆ ((𝐴𝑜 𝐶) ·𝑜 𝐷))
28 omcl 7503 . . . . . . . . . . . . . . . 16 (((𝐴𝑜 𝐶) ∈ On ∧ 𝐷 ∈ On) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ On)
296, 16, 28syl2anc 691 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ On)
30 simplrl 796 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐸 ∈ (𝐴𝑜 𝐶))
31 onelon 5665 . . . . . . . . . . . . . . . 16 (((𝐴𝑜 𝐶) ∈ On ∧ 𝐸 ∈ (𝐴𝑜 𝐶)) → 𝐸 ∈ On)
326, 30, 31syl2anc 691 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐸 ∈ On)
33 oaword1 7519 . . . . . . . . . . . . . . 15 ((((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ⊆ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸))
3429, 32, 33syl2anc 691 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ⊆ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸))
35 simplrr 797 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)
3634, 35sseqtrd 3604 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ⊆ 𝐵)
3727, 36sstrd 3578 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴𝑜 𝐶) ⊆ 𝐵)
38 oeeu.1 . . . . . . . . . . . . . . 15 𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}
3938oeeulem 7568 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝑋 ∈ On ∧ (𝐴𝑜 𝑋) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc 𝑋)))
4039simp3d 1068 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ (𝐴𝑜 suc 𝑋))
4140ad2antrr 758 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐵 ∈ (𝐴𝑜 suc 𝑋))
4239simp1d 1066 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ On)
4342ad2antrr 758 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝑋 ∈ On)
44 suceloni 6905 . . . . . . . . . . . . . . 15 (𝑋 ∈ On → suc 𝑋 ∈ On)
4543, 44syl 17 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → suc 𝑋 ∈ On)
46 oecl 7504 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ suc 𝑋 ∈ On) → (𝐴𝑜 suc 𝑋) ∈ On)
473, 45, 46syl2anc 691 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴𝑜 suc 𝑋) ∈ On)
48 ontr2 5689 . . . . . . . . . . . . 13 (((𝐴𝑜 𝐶) ∈ On ∧ (𝐴𝑜 suc 𝑋) ∈ On) → (((𝐴𝑜 𝐶) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc 𝑋)) → (𝐴𝑜 𝐶) ∈ (𝐴𝑜 suc 𝑋)))
496, 47, 48syl2anc 691 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (((𝐴𝑜 𝐶) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc 𝑋)) → (𝐴𝑜 𝐶) ∈ (𝐴𝑜 suc 𝑋)))
5037, 41, 49mp2and 711 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴𝑜 𝐶) ∈ (𝐴𝑜 suc 𝑋))
51 simplll 794 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐴 ∈ (On ∖ 2𝑜))
52 oeord 7555 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ suc 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝐶 ∈ suc 𝑋 ↔ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 suc 𝑋)))
534, 45, 51, 52syl3anc 1318 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐶 ∈ suc 𝑋 ↔ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 suc 𝑋)))
5450, 53mpbird 246 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐶 ∈ suc 𝑋)
55 onsssuc 5730 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝑋 ∈ On) → (𝐶𝑋𝐶 ∈ suc 𝑋))
564, 43, 55syl2anc 691 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐶𝑋𝐶 ∈ suc 𝑋))
5754, 56mpbird 246 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐶𝑋)
5839simp2d 1067 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴𝑜 𝑋) ⊆ 𝐵)
5958ad2antrr 758 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴𝑜 𝑋) ⊆ 𝐵)
60 eloni 5650 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → Ord 𝐴)
613, 60syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → Ord 𝐴)
62 ordsucss 6910 . . . . . . . . . . . . . . . 16 (Ord 𝐴 → (𝐷𝐴 → suc 𝐷𝐴))
6361, 14, 62sylc 63 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → suc 𝐷𝐴)
64 suceloni 6905 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ On → suc 𝐷 ∈ On)
6516, 64syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → suc 𝐷 ∈ On)
66 dif20el 7472 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
6751, 66syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ∅ ∈ 𝐴)
68 oen0 7553 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝐶))
693, 4, 67, 68syl21anc 1317 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ∅ ∈ (𝐴𝑜 𝐶))
70 omword 7537 . . . . . . . . . . . . . . . 16 (((suc 𝐷 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴𝑜 𝐶) ∈ On) ∧ ∅ ∈ (𝐴𝑜 𝐶)) → (suc 𝐷𝐴 ↔ ((𝐴𝑜 𝐶) ·𝑜 suc 𝐷) ⊆ ((𝐴𝑜 𝐶) ·𝑜 𝐴)))
7165, 3, 6, 69, 70syl31anc 1321 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (suc 𝐷𝐴 ↔ ((𝐴𝑜 𝐶) ·𝑜 suc 𝐷) ⊆ ((𝐴𝑜 𝐶) ·𝑜 𝐴)))
7263, 71mpbid 221 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴𝑜 𝐶) ·𝑜 suc 𝐷) ⊆ ((𝐴𝑜 𝐶) ·𝑜 𝐴))
73 oaord 7514 . . . . . . . . . . . . . . . . . 18 ((𝐸 ∈ On ∧ (𝐴𝑜 𝐶) ∈ On ∧ ((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ On) → (𝐸 ∈ (𝐴𝑜 𝐶) ↔ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴𝑜 𝐶))))
7432, 6, 29, 73syl3anc 1318 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐸 ∈ (𝐴𝑜 𝐶) ↔ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴𝑜 𝐶))))
7530, 74mpbid 221 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴𝑜 𝐶)))
7635, 75eqeltrrd 2689 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐵 ∈ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴𝑜 𝐶)))
77 odi 7546 . . . . . . . . . . . . . . . . 17 (((𝐴𝑜 𝐶) ∈ On ∧ 𝐷 ∈ On ∧ 1𝑜 ∈ On) → ((𝐴𝑜 𝐶) ·𝑜 (𝐷 +𝑜 1𝑜)) = (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 ((𝐴𝑜 𝐶) ·𝑜 1𝑜)))
786, 16, 23, 77syl3anc 1318 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴𝑜 𝐶) ·𝑜 (𝐷 +𝑜 1𝑜)) = (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 ((𝐴𝑜 𝐶) ·𝑜 1𝑜)))
79 oa1suc 7498 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ On → (𝐷 +𝑜 1𝑜) = suc 𝐷)
8016, 79syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐷 +𝑜 1𝑜) = suc 𝐷)
8180oveq2d 6565 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴𝑜 𝐶) ·𝑜 (𝐷 +𝑜 1𝑜)) = ((𝐴𝑜 𝐶) ·𝑜 suc 𝐷))
828oveq2d 6565 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 ((𝐴𝑜 𝐶) ·𝑜 1𝑜)) = (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴𝑜 𝐶)))
8378, 81, 823eqtr3d 2652 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴𝑜 𝐶) ·𝑜 suc 𝐷) = (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴𝑜 𝐶)))
8476, 83eleqtrrd 2691 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐵 ∈ ((𝐴𝑜 𝐶) ·𝑜 suc 𝐷))
8572, 84sseldd 3569 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐵 ∈ ((𝐴𝑜 𝐶) ·𝑜 𝐴))
86 oesuc 7494 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) = ((𝐴𝑜 𝐶) ·𝑜 𝐴))
873, 4, 86syl2anc 691 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴𝑜 suc 𝐶) = ((𝐴𝑜 𝐶) ·𝑜 𝐴))
8885, 87eleqtrrd 2691 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐵 ∈ (𝐴𝑜 suc 𝐶))
89 oecl 7504 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
903, 43, 89syl2anc 691 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴𝑜 𝑋) ∈ On)
91 suceloni 6905 . . . . . . . . . . . . . . 15 (𝐶 ∈ On → suc 𝐶 ∈ On)
9291ad2antrl 760 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → suc 𝐶 ∈ On)
93 oecl 7504 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) ∈ On)
943, 92, 93syl2anc 691 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴𝑜 suc 𝐶) ∈ On)
95 ontr2 5689 . . . . . . . . . . . . 13 (((𝐴𝑜 𝑋) ∈ On ∧ (𝐴𝑜 suc 𝐶) ∈ On) → (((𝐴𝑜 𝑋) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc 𝐶)) → (𝐴𝑜 𝑋) ∈ (𝐴𝑜 suc 𝐶)))
9690, 94, 95syl2anc 691 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (((𝐴𝑜 𝑋) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc 𝐶)) → (𝐴𝑜 𝑋) ∈ (𝐴𝑜 suc 𝐶)))
9759, 88, 96mp2and 711 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴𝑜 𝑋) ∈ (𝐴𝑜 suc 𝐶))
98 oeord 7555 . . . . . . . . . . . 12 ((𝑋 ∈ On ∧ suc 𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑋 ∈ suc 𝐶 ↔ (𝐴𝑜 𝑋) ∈ (𝐴𝑜 suc 𝐶)))
9943, 92, 51, 98syl3anc 1318 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝑋 ∈ suc 𝐶 ↔ (𝐴𝑜 𝑋) ∈ (𝐴𝑜 suc 𝐶)))
10097, 99mpbird 246 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝑋 ∈ suc 𝐶)
101 onsssuc 5730 . . . . . . . . . . 11 ((𝑋 ∈ On ∧ 𝐶 ∈ On) → (𝑋𝐶𝑋 ∈ suc 𝐶))
10243, 4, 101syl2anc 691 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝑋𝐶𝑋 ∈ suc 𝐶))
103100, 102mpbird 246 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝑋𝐶)
10457, 103eqssd 3585 . . . . . . . 8 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐶 = 𝑋)
105104, 16jca 553 . . . . . . 7 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐶 = 𝑋𝐷 ∈ On))
106 simprl 790 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐶 = 𝑋)
10742ad2antrr 758 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝑋 ∈ On)
108106, 107eqeltrd 2688 . . . . . . . 8 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐶 ∈ On)
1092ad2antrr 758 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐴 ∈ On)
110109, 108, 5syl2anc 691 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴𝑜 𝐶) ∈ On)
111 simprr 792 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐷 ∈ On)
112110, 111, 28syl2anc 691 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ On)
113 simplrl 796 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐸 ∈ (𝐴𝑜 𝐶))
114110, 113, 31syl2anc 691 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐸 ∈ On)
115112, 114, 33syl2anc 691 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ⊆ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸))
116 simplrr 797 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)
117115, 116sseqtrd 3604 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ⊆ 𝐵)
11840ad2antrr 758 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐵 ∈ (𝐴𝑜 suc 𝑋))
119 suceq 5707 . . . . . . . . . . . . . . 15 (𝐶 = 𝑋 → suc 𝐶 = suc 𝑋)
120119ad2antrl 760 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → suc 𝐶 = suc 𝑋)
121120oveq2d 6565 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴𝑜 suc 𝐶) = (𝐴𝑜 suc 𝑋))
122109, 108, 86syl2anc 691 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴𝑜 suc 𝐶) = ((𝐴𝑜 𝐶) ·𝑜 𝐴))
123121, 122eqtr3d 2646 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴𝑜 suc 𝑋) = ((𝐴𝑜 𝐶) ·𝑜 𝐴))
124118, 123eleqtrd 2690 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐵 ∈ ((𝐴𝑜 𝐶) ·𝑜 𝐴))
125 omcl 7503 . . . . . . . . . . . . 13 (((𝐴𝑜 𝐶) ∈ On ∧ 𝐴 ∈ On) → ((𝐴𝑜 𝐶) ·𝑜 𝐴) ∈ On)
126110, 109, 125syl2anc 691 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴𝑜 𝐶) ·𝑜 𝐴) ∈ On)
127 ontr2 5689 . . . . . . . . . . . 12 ((((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ On ∧ ((𝐴𝑜 𝐶) ·𝑜 𝐴) ∈ On) → ((((𝐴𝑜 𝐶) ·𝑜 𝐷) ⊆ 𝐵𝐵 ∈ ((𝐴𝑜 𝐶) ·𝑜 𝐴)) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ ((𝐴𝑜 𝐶) ·𝑜 𝐴)))
128112, 126, 127syl2anc 691 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((((𝐴𝑜 𝐶) ·𝑜 𝐷) ⊆ 𝐵𝐵 ∈ ((𝐴𝑜 𝐶) ·𝑜 𝐴)) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ ((𝐴𝑜 𝐶) ·𝑜 𝐴)))
129117, 124, 128mp2and 711 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ ((𝐴𝑜 𝐶) ·𝑜 𝐴))
13066adantr 480 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∅ ∈ 𝐴)
131130ad2antrr 758 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ∅ ∈ 𝐴)
132109, 108, 131, 68syl21anc 1317 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ∅ ∈ (𝐴𝑜 𝐶))
133 omord2 7534 . . . . . . . . . . 11 (((𝐷 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴𝑜 𝐶) ∈ On) ∧ ∅ ∈ (𝐴𝑜 𝐶)) → (𝐷𝐴 ↔ ((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ ((𝐴𝑜 𝐶) ·𝑜 𝐴)))
134111, 109, 110, 132, 133syl31anc 1321 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐷𝐴 ↔ ((𝐴𝑜 𝐶) ·𝑜 𝐷) ∈ ((𝐴𝑜 𝐶) ·𝑜 𝐴)))
135129, 134mpbird 246 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐷𝐴)
136106oveq2d 6565 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴𝑜 𝐶) = (𝐴𝑜 𝑋))
13758ad2antrr 758 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴𝑜 𝑋) ⊆ 𝐵)
138136, 137eqsstrd 3602 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴𝑜 𝐶) ⊆ 𝐵)
139 eldifi 3694 . . . . . . . . . . . . . 14 (𝐵 ∈ (On ∖ 1𝑜) → 𝐵 ∈ On)
140139adantl 481 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ On)
141140ad2antrr 758 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐵 ∈ On)
142 ontri1 5674 . . . . . . . . . . . 12 (((𝐴𝑜 𝐶) ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝑜 𝐶) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴𝑜 𝐶)))
143110, 141, 142syl2anc 691 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴𝑜 𝐶) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴𝑜 𝐶)))
144138, 143mpbid 221 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ¬ 𝐵 ∈ (𝐴𝑜 𝐶))
145 om0 7484 . . . . . . . . . . . . . . . . 17 ((𝐴𝑜 𝐶) ∈ On → ((𝐴𝑜 𝐶) ·𝑜 ∅) = ∅)
146110, 145syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴𝑜 𝐶) ·𝑜 ∅) = ∅)
147146oveq1d 6564 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (((𝐴𝑜 𝐶) ·𝑜 ∅) +𝑜 𝐸) = (∅ +𝑜 𝐸))
148 oa0r 7505 . . . . . . . . . . . . . . . 16 (𝐸 ∈ On → (∅ +𝑜 𝐸) = 𝐸)
149114, 148syl 17 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (∅ +𝑜 𝐸) = 𝐸)
150147, 149eqtrd 2644 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (((𝐴𝑜 𝐶) ·𝑜 ∅) +𝑜 𝐸) = 𝐸)
151150, 113eqeltrd 2688 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (((𝐴𝑜 𝐶) ·𝑜 ∅) +𝑜 𝐸) ∈ (𝐴𝑜 𝐶))
152 oveq2 6557 . . . . . . . . . . . . . . 15 (𝐷 = ∅ → ((𝐴𝑜 𝐶) ·𝑜 𝐷) = ((𝐴𝑜 𝐶) ·𝑜 ∅))
153152oveq1d 6564 . . . . . . . . . . . . . 14 (𝐷 = ∅ → (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = (((𝐴𝑜 𝐶) ·𝑜 ∅) +𝑜 𝐸))
154153eleq1d 2672 . . . . . . . . . . . . 13 (𝐷 = ∅ → ((((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (𝐴𝑜 𝐶) ↔ (((𝐴𝑜 𝐶) ·𝑜 ∅) +𝑜 𝐸) ∈ (𝐴𝑜 𝐶)))
155151, 154syl5ibrcom 236 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐷 = ∅ → (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (𝐴𝑜 𝐶)))
156116eleq1d 2672 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (𝐴𝑜 𝐶) ↔ 𝐵 ∈ (𝐴𝑜 𝐶)))
157155, 156sylibd 228 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐷 = ∅ → 𝐵 ∈ (𝐴𝑜 𝐶)))
158157necon3bd 2796 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (¬ 𝐵 ∈ (𝐴𝑜 𝐶) → 𝐷 ≠ ∅))
159144, 158mpd 15 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐷 ≠ ∅)
160135, 159, 10sylanbrc 695 . . . . . . . 8 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐷 ∈ (𝐴 ∖ 1𝑜))
161108, 160jca 553 . . . . . . 7 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)))
162105, 161impbida 873 . . . . . 6 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) → ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ↔ (𝐶 = 𝑋𝐷 ∈ On)))
163162ex 449 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ((𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) → ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ↔ (𝐶 = 𝑋𝐷 ∈ On))))
164163pm5.32rd 670 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ ((𝐶 = 𝑋𝐷 ∈ On) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵))))
165 anass 679 . . . 4 (((𝐶 = 𝑋𝐷 ∈ On) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ (𝐶 = 𝑋 ∧ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵))))
166164, 165syl6bb 275 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ (𝐶 = 𝑋 ∧ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)))))
167 3anass 1035 . . . . . 6 ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)))
168 oveq2 6557 . . . . . . . 8 (𝐶 = 𝑋 → (𝐴𝑜 𝐶) = (𝐴𝑜 𝑋))
169168eleq2d 2673 . . . . . . 7 (𝐶 = 𝑋 → (𝐸 ∈ (𝐴𝑜 𝐶) ↔ 𝐸 ∈ (𝐴𝑜 𝑋)))
170168oveq1d 6564 . . . . . . . . 9 (𝐶 = 𝑋 → ((𝐴𝑜 𝐶) ·𝑜 𝐷) = ((𝐴𝑜 𝑋) ·𝑜 𝐷))
171170oveq1d 6564 . . . . . . . 8 (𝐶 = 𝑋 → (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸))
172171eqeq1d 2612 . . . . . . 7 (𝐶 = 𝑋 → ((((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵 ↔ (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵))
173169, 1723anbi23d 1394 . . . . . 6 (𝐶 = 𝑋 → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐷 ∈ On ∧ 𝐸 ∈ (𝐴𝑜 𝑋) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)))
174167, 173syl5bbr 273 . . . . 5 (𝐶 = 𝑋 → ((𝐷 ∈ On ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ (𝐷 ∈ On ∧ 𝐸 ∈ (𝐴𝑜 𝑋) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)))
1752, 42, 89syl2anc 691 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴𝑜 𝑋) ∈ On)
176 oen0 7553 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝑋 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝑋))
1772, 42, 130, 176syl21anc 1317 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∅ ∈ (𝐴𝑜 𝑋))
178 ne0i 3880 . . . . . . 7 (∅ ∈ (𝐴𝑜 𝑋) → (𝐴𝑜 𝑋) ≠ ∅)
179177, 178syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴𝑜 𝑋) ≠ ∅)
180 omeu 7552 . . . . . . 7 (((𝐴𝑜 𝑋) ∈ On ∧ 𝐵 ∈ On ∧ (𝐴𝑜 𝑋) ≠ ∅) → ∃!𝑎𝑑 ∈ On ∃𝑒 ∈ (𝐴𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵))
181 oeeu.2 . . . . . . . . 9 𝑃 = (℩𝑤𝑦 ∈ On ∃𝑧 ∈ (𝐴𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
182 opeq1 4340 . . . . . . . . . . . . . 14 (𝑦 = 𝑑 → ⟨𝑦, 𝑧⟩ = ⟨𝑑, 𝑧⟩)
183182eqeq2d 2620 . . . . . . . . . . . . 13 (𝑦 = 𝑑 → (𝑤 = ⟨𝑦, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑧⟩))
184 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑦 = 𝑑 → ((𝐴𝑜 𝑋) ·𝑜 𝑦) = ((𝐴𝑜 𝑋) ·𝑜 𝑑))
185184oveq1d 6564 . . . . . . . . . . . . . 14 (𝑦 = 𝑑 → (((𝐴𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧))
186185eqeq1d 2612 . . . . . . . . . . . . 13 (𝑦 = 𝑑 → ((((𝐴𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵 ↔ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧) = 𝐵))
187183, 186anbi12d 743 . . . . . . . . . . . 12 (𝑦 = 𝑑 → ((𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ (𝑤 = ⟨𝑑, 𝑧⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧) = 𝐵)))
188 opeq2 4341 . . . . . . . . . . . . . 14 (𝑧 = 𝑒 → ⟨𝑑, 𝑧⟩ = ⟨𝑑, 𝑒⟩)
189188eqeq2d 2620 . . . . . . . . . . . . 13 (𝑧 = 𝑒 → (𝑤 = ⟨𝑑, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑒⟩))
190 oveq2 6557 . . . . . . . . . . . . . 14 (𝑧 = 𝑒 → (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧) = (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒))
191190eqeq1d 2612 . . . . . . . . . . . . 13 (𝑧 = 𝑒 → ((((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧) = 𝐵 ↔ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵))
192189, 191anbi12d 743 . . . . . . . . . . . 12 (𝑧 = 𝑒 → ((𝑤 = ⟨𝑑, 𝑧⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧) = 𝐵) ↔ (𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵)))
193187, 192cbvrex2v 3156 . . . . . . . . . . 11 (∃𝑦 ∈ On ∃𝑧 ∈ (𝐴𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ∃𝑑 ∈ On ∃𝑒 ∈ (𝐴𝑜 𝑋)(𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵))
194 eqeq1 2614 . . . . . . . . . . . . 13 (𝑤 = 𝑎 → (𝑤 = ⟨𝑑, 𝑒⟩ ↔ 𝑎 = ⟨𝑑, 𝑒⟩))
195194anbi1d 737 . . . . . . . . . . . 12 (𝑤 = 𝑎 → ((𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵) ↔ (𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵)))
1961952rexbidv 3039 . . . . . . . . . . 11 (𝑤 = 𝑎 → (∃𝑑 ∈ On ∃𝑒 ∈ (𝐴𝑜 𝑋)(𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵) ↔ ∃𝑑 ∈ On ∃𝑒 ∈ (𝐴𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵)))
197193, 196syl5bb 271 . . . . . . . . . 10 (𝑤 = 𝑎 → (∃𝑦 ∈ On ∃𝑧 ∈ (𝐴𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ∃𝑑 ∈ On ∃𝑒 ∈ (𝐴𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵)))
198197cbviotav 5774 . . . . . . . . 9 (℩𝑤𝑦 ∈ On ∃𝑧 ∈ (𝐴𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) = (℩𝑎𝑑 ∈ On ∃𝑒 ∈ (𝐴𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵))
199181, 198eqtri 2632 . . . . . . . 8 𝑃 = (℩𝑎𝑑 ∈ On ∃𝑒 ∈ (𝐴𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵))
200 oeeu.3 . . . . . . . 8 𝑌 = (1st𝑃)
201 oeeu.4 . . . . . . . 8 𝑍 = (2nd𝑃)
202 oveq2 6557 . . . . . . . . . 10 (𝑑 = 𝐷 → ((𝐴𝑜 𝑋) ·𝑜 𝑑) = ((𝐴𝑜 𝑋) ·𝑜 𝐷))
203202oveq1d 6564 . . . . . . . . 9 (𝑑 = 𝐷 → (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝑒))
204203eqeq1d 2612 . . . . . . . 8 (𝑑 = 𝐷 → ((((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵 ↔ (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝑒) = 𝐵))
205 oveq2 6557 . . . . . . . . 9 (𝑒 = 𝐸 → (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝑒) = (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸))
206205eqeq1d 2612 . . . . . . . 8 (𝑒 = 𝐸 → ((((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝑒) = 𝐵 ↔ (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵))
207199, 200, 201, 204, 206opiota 7118 . . . . . . 7 (∃!𝑎𝑑 ∈ On ∃𝑒 ∈ (𝐴𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵) → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴𝑜 𝑋) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐷 = 𝑌𝐸 = 𝑍)))
208180, 207syl 17 . . . . . 6 (((𝐴𝑜 𝑋) ∈ On ∧ 𝐵 ∈ On ∧ (𝐴𝑜 𝑋) ≠ ∅) → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴𝑜 𝑋) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐷 = 𝑌𝐸 = 𝑍)))
209175, 140, 179, 208syl3anc 1318 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴𝑜 𝑋) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐷 = 𝑌𝐸 = 𝑍)))
210174, 209sylan9bbr 733 . . . 4 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ 𝐶 = 𝑋) → ((𝐷 ∈ On ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ (𝐷 = 𝑌𝐸 = 𝑍)))
211210pm5.32da 671 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ((𝐶 = 𝑋 ∧ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵))) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌𝐸 = 𝑍))))
212166, 211bitrd 267 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌𝐸 = 𝑍))))
213 3an4anass 1283 . 2 (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜) ∧ 𝐸 ∈ (𝐴𝑜 𝐶)) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴𝑜 𝐶) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)))
214 3anass 1035 . 2 ((𝐶 = 𝑋𝐷 = 𝑌𝐸 = 𝑍) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌𝐸 = 𝑍)))
215212, 213, 2143bitr4g 302 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜) ∧ 𝐸 ∈ (𝐴𝑜 𝐶)) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐶 = 𝑋𝐷 = 𝑌𝐸 = 𝑍)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∩ cint 4410  Ord word 5639  Oncon0 5640  suc csuc 5642  ℩cio 5766  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  1𝑜c1o 7440  2𝑜c2o 7441   +𝑜 coa 7444   ·𝑜 comu 7445   ↑𝑜 coe 7446 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453 This theorem is referenced by:  oeeu  7570  cantnflem3  8471  cantnflem4  8472
 Copyright terms: Public domain W3C validator