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Theorem oeeu 7570
 Description: The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)
Assertion
Ref Expression
oeeu ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧

Proof of Theorem oeeu
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . . 6 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}
21oeeulem 7568 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ( {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ On ∧ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})))
32simp1d 1066 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ On)
4 elex 3185 . . . 4 ( {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ On → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ V)
53, 4syl 17 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ V)
6 fvex 6113 . . . 4 (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V
76a1i 11 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V)
8 fvex 6113 . . . 4 (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V
98a1i 11 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V)
10 eqid 2610 . . . 4 (℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)) = (℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))
11 eqid 2610 . . . 4 (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) = (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)))
12 eqid 2610 . . . 4 (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) = (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)))
131, 10, 11, 12oeeui 7569 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ (𝑥 = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∧ 𝑦 = (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∧ 𝑧 = (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))))))
145, 7, 9, 13euotd 4900 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
15 df-3an 1033 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)))
16 ancom 465 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))))
1715, 16bitri 263 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))))
1817anbi1i 727 . . . . . . . . 9 (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
1918anbi2i 726 . . . . . . . 8 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
20 an12 834 . . . . . . . 8 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
21 anass 679 . . . . . . . 8 (((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
2219, 20, 213bitri 285 . . . . . . 7 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
2322exbii 1764 . . . . . 6 (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
24 df-rex 2902 . . . . . 6 (∃𝑧 ∈ (𝐴𝑜 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
25 r19.42v 3073 . . . . . 6 (∃𝑧 ∈ (𝐴𝑜 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
2623, 24, 253bitr2i 287 . . . . 5 (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
27262exbii 1765 . . . 4 (∃𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑥𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
28 r2ex 3043 . . . 4 (∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ∃𝑥𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
2927, 28bitr4i 266 . . 3 (∃𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
3029eubii 2480 . 2 (∃!𝑤𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
3114, 30sylib 207 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ⟨cop 4131  ⟨cotp 4133  ∪ cuni 4372  ∩ cint 4410  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-ot 4134  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: (None)
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