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

Theorem oeordi 7554
 Description: Ordering law for ordinal exponentiation. Proposition 8.33 of [TakeutiZaring] p. 67. (Contributed by NM, 5-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oeordi ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))

Proof of Theorem oeordi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6557 . . . . 5 (𝑥 = suc 𝐴 → (𝐶𝑜 𝑥) = (𝐶𝑜 suc 𝐴))
21eleq2d 2673 . . . 4 (𝑥 = suc 𝐴 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
32imbi2d 329 . . 3 (𝑥 = suc 𝐴 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
4 oveq2 6557 . . . . 5 (𝑥 = 𝑦 → (𝐶𝑜 𝑥) = (𝐶𝑜 𝑦))
54eleq2d 2673 . . . 4 (𝑥 = 𝑦 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)))
65imbi2d 329 . . 3 (𝑥 = 𝑦 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
7 oveq2 6557 . . . . 5 (𝑥 = suc 𝑦 → (𝐶𝑜 𝑥) = (𝐶𝑜 suc 𝑦))
87eleq2d 2673 . . . 4 (𝑥 = suc 𝑦 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦)))
98imbi2d 329 . . 3 (𝑥 = suc 𝑦 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦))))
10 oveq2 6557 . . . . 5 (𝑥 = 𝐵 → (𝐶𝑜 𝑥) = (𝐶𝑜 𝐵))
1110eleq2d 2673 . . . 4 (𝑥 = 𝐵 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
1211imbi2d 329 . . 3 (𝑥 = 𝐵 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵))))
13 eldifi 3694 . . . . . . . 8 (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
14 oecl 7504 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
1513, 14sylan 487 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
16 om1 7509 . . . . . . 7 ((𝐶𝑜 𝐴) ∈ On → ((𝐶𝑜 𝐴) ·𝑜 1𝑜) = (𝐶𝑜 𝐴))
1715, 16syl 17 . . . . . 6 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ((𝐶𝑜 𝐴) ·𝑜 1𝑜) = (𝐶𝑜 𝐴))
18 ondif2 7469 . . . . . . . . 9 (𝐶 ∈ (On ∖ 2𝑜) ↔ (𝐶 ∈ On ∧ 1𝑜𝐶))
1918simprbi 479 . . . . . . . 8 (𝐶 ∈ (On ∖ 2𝑜) → 1𝑜𝐶)
2019adantr 480 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 1𝑜𝐶)
2113adantr 480 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐶 ∈ On)
22 simpr 476 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
23 dif20el 7472 . . . . . . . . . 10 (𝐶 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐶)
2423adantr 480 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25 oen0 7553 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶𝑜 𝐴))
2621, 22, 24, 25syl21anc 1317 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶𝑜 𝐴))
27 omordi 7533 . . . . . . . 8 (((𝐶 ∈ On ∧ (𝐶𝑜 𝐴) ∈ On) ∧ ∅ ∈ (𝐶𝑜 𝐴)) → (1𝑜𝐶 → ((𝐶𝑜 𝐴) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶)))
2821, 15, 26, 27syl21anc 1317 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (1𝑜𝐶 → ((𝐶𝑜 𝐴) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶)))
2920, 28mpd 15 . . . . . 6 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ((𝐶𝑜 𝐴) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶))
3017, 29eqeltrrd 2689 . . . . 5 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶))
31 oesuc 7494 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 suc 𝐴) = ((𝐶𝑜 𝐴) ·𝑜 𝐶))
3213, 31sylan 487 . . . . 5 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 suc 𝐴) = ((𝐶𝑜 𝐴) ·𝑜 𝐶))
3330, 32eleqtrrd 2691 . . . 4 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))
3433expcom 450 . . 3 (𝐴 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
35 oecl 7504 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ On)
3613, 35sylan 487 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ On)
37 om1 7509 . . . . . . . . . 10 ((𝐶𝑜 𝑦) ∈ On → ((𝐶𝑜 𝑦) ·𝑜 1𝑜) = (𝐶𝑜 𝑦))
3836, 37syl 17 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ((𝐶𝑜 𝑦) ·𝑜 1𝑜) = (𝐶𝑜 𝑦))
3919adantr 480 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 1𝑜𝐶)
4013adantr 480 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
41 simpr 476 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝑦 ∈ On)
4223adantr 480 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ 𝐶)
43 oen0 7553 . . . . . . . . . . . 12 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶𝑜 𝑦))
4440, 41, 42, 43syl21anc 1317 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶𝑜 𝑦))
45 omordi 7533 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝐶𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐶𝑜 𝑦)) → (1𝑜𝐶 → ((𝐶𝑜 𝑦) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶)))
4640, 36, 44, 45syl21anc 1317 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (1𝑜𝐶 → ((𝐶𝑜 𝑦) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶)))
4739, 46mpd 15 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ((𝐶𝑜 𝑦) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶))
4838, 47eqeltrrd 2689 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶))
49 oesuc 7494 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) = ((𝐶𝑜 𝑦) ·𝑜 𝐶))
5013, 49sylan 487 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) = ((𝐶𝑜 𝑦) ·𝑜 𝐶))
5148, 50eleqtrrd 2691 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ (𝐶𝑜 suc 𝑦))
52 suceloni 6905 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
53 oecl 7504 . . . . . . . . 9 ((𝐶 ∈ On ∧ suc 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) ∈ On)
5413, 52, 53syl2an 493 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) ∈ On)
55 ontr1 5688 . . . . . . . 8 ((𝐶𝑜 suc 𝑦) ∈ On → (((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) ∧ (𝐶𝑜 𝑦) ∈ (𝐶𝑜 suc 𝑦)) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦)))
5654, 55syl 17 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) ∧ (𝐶𝑜 𝑦) ∈ (𝐶𝑜 suc 𝑦)) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦)))
5751, 56mpan2d 706 . . . . . 6 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦)))
5857expcom 450 . . . . 5 (𝑦 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦))))
5958adantr 480 . . . 4 ((𝑦 ∈ On ∧ 𝐴𝑦) → (𝐶 ∈ (On ∖ 2𝑜) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦))))
6059a2d 29 . . 3 ((𝑦 ∈ On ∧ 𝐴𝑦) → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦))))
61 bi2.04 375 . . . . . 6 ((𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
6261ralbii 2963 . . . . 5 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ ∀𝑦𝑥 (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
63 r19.21v 2943 . . . . 5 (∀𝑦𝑥 (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
6462, 63bitri 263 . . . 4 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
65 limsuc 6941 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
6665biimpa 500 . . . . . . . . 9 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
67 elex 3185 . . . . . . . . . . . . 13 (suc 𝐴𝑥 → suc 𝐴 ∈ V)
68 sucexb 6901 . . . . . . . . . . . . . 14 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69 sucidg 5720 . . . . . . . . . . . . . 14 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7068, 69sylbir 224 . . . . . . . . . . . . 13 (suc 𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7167, 70syl 17 . . . . . . . . . . . 12 (suc 𝐴𝑥𝐴 ∈ suc 𝐴)
72 eleq2 2677 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → (𝐴𝑦𝐴 ∈ suc 𝐴))
73 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑦 = suc 𝐴 → (𝐶𝑜 𝑦) = (𝐶𝑜 suc 𝐴))
7473eleq2d 2673 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
7572, 74imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = suc 𝐴 → ((𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) ↔ (𝐴 ∈ suc 𝐴 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7675rspcv 3278 . . . . . . . . . . . 12 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐴 ∈ suc 𝐴 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7771, 76mpid 43 . . . . . . . . . . 11 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
7877anc2li 578 . . . . . . . . . 10 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (suc 𝐴𝑥 ∧ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7973eliuni 4462 . . . . . . . . . 10 ((suc 𝐴𝑥 ∧ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)) → (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦))
8078, 79syl6 34 . . . . . . . . 9 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦)))
8166, 80syl 17 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦)))
8281adantr 480 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦)))
8313adantl 481 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → 𝐶 ∈ On)
84 simpl 472 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → Lim 𝑥)
8523adantl 481 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → ∅ ∈ 𝐶)
86 vex 3176 . . . . . . . . . . 11 𝑥 ∈ V
87 oelim 7501 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
8886, 87mpanlr1 718 . . . . . . . . . 10 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
8983, 84, 85, 88syl21anc 1317 . . . . . . . . 9 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
9089adantlr 747 . . . . . . . 8 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
9190eleq2d 2673 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦)))
9282, 91sylibrd 248 . . . . . 6 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)))
9392ex 449 . . . . 5 ((Lim 𝑥𝐴𝑥) → (𝐶 ∈ (On ∖ 2𝑜) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥))))
9493a2d 29 . . . 4 ((Lim 𝑥𝐴𝑥) → ((𝐶 ∈ (On ∖ 2𝑜) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥))))
9564, 94syl5bi 231 . . 3 ((Lim 𝑥𝐴𝑥) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥))))
963, 6, 9, 12, 34, 60, 95tfindsg2 6953 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
9796impancom 455 1 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  ∪ ciun 4455  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1𝑜c1o 7440  2𝑜c2o 7441   ·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-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-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-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:  oeord  7555  oecan  7556  oeworde  7560  oelimcl  7567
 Copyright terms: Public domain W3C validator