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Theorem oeordi 7531
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 6535 . . . . 5 (𝑥 = suc 𝐴 → (𝐶𝑜 𝑥) = (𝐶𝑜 suc 𝐴))
21eleq2d 2672 . . . 4 (𝑥 = suc 𝐴 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
32imbi2d 328 . . 3 (𝑥 = suc 𝐴 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
4 oveq2 6535 . . . . 5 (𝑥 = 𝑦 → (𝐶𝑜 𝑥) = (𝐶𝑜 𝑦))
54eleq2d 2672 . . . 4 (𝑥 = 𝑦 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)))
65imbi2d 328 . . 3 (𝑥 = 𝑦 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
7 oveq2 6535 . . . . 5 (𝑥 = suc 𝑦 → (𝐶𝑜 𝑥) = (𝐶𝑜 suc 𝑦))
87eleq2d 2672 . . . 4 (𝑥 = suc 𝑦 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦)))
98imbi2d 328 . . 3 (𝑥 = suc 𝑦 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦))))
10 oveq2 6535 . . . . 5 (𝑥 = 𝐵 → (𝐶𝑜 𝑥) = (𝐶𝑜 𝐵))
1110eleq2d 2672 . . . 4 (𝑥 = 𝐵 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
1211imbi2d 328 . . 3 (𝑥 = 𝐵 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵))))
13 eldifi 3693 . . . . . . . 8 (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
14 oecl 7481 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
1513, 14sylan 486 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
16 om1 7486 . . . . . . 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 7446 . . . . . . . . 9 (𝐶 ∈ (On ∖ 2𝑜) ↔ (𝐶 ∈ On ∧ 1𝑜𝐶))
1918simprbi 478 . . . . . . . 8 (𝐶 ∈ (On ∖ 2𝑜) → 1𝑜𝐶)
2019adantr 479 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 1𝑜𝐶)
2113adantr 479 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐶 ∈ On)
22 simpr 475 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
23 dif20el 7449 . . . . . . . . . 10 (𝐶 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐶)
2423adantr 479 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25 oen0 7530 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶𝑜 𝐴))
2621, 22, 24, 25syl21anc 1316 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶𝑜 𝐴))
27 omordi 7510 . . . . . . . 8 (((𝐶 ∈ On ∧ (𝐶𝑜 𝐴) ∈ On) ∧ ∅ ∈ (𝐶𝑜 𝐴)) → (1𝑜𝐶 → ((𝐶𝑜 𝐴) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶)))
2821, 15, 26, 27syl21anc 1316 . . . . . . 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 2688 . . . . 5 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶))
31 oesuc 7471 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 suc 𝐴) = ((𝐶𝑜 𝐴) ·𝑜 𝐶))
3213, 31sylan 486 . . . . 5 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 suc 𝐴) = ((𝐶𝑜 𝐴) ·𝑜 𝐶))
3330, 32eleqtrrd 2690 . . . 4 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))
3433expcom 449 . . 3 (𝐴 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
35 oecl 7481 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ On)
3613, 35sylan 486 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ On)
37 om1 7486 . . . . . . . . . 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 479 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 1𝑜𝐶)
4013adantr 479 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
41 simpr 475 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝑦 ∈ On)
4223adantr 479 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ 𝐶)
43 oen0 7530 . . . . . . . . . . . 12 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶𝑜 𝑦))
4440, 41, 42, 43syl21anc 1316 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶𝑜 𝑦))
45 omordi 7510 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝐶𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐶𝑜 𝑦)) → (1𝑜𝐶 → ((𝐶𝑜 𝑦) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶)))
4640, 36, 44, 45syl21anc 1316 . . . . . . . . . 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 2688 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶))
49 oesuc 7471 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) = ((𝐶𝑜 𝑦) ·𝑜 𝐶))
5013, 49sylan 486 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) = ((𝐶𝑜 𝑦) ·𝑜 𝐶))
5148, 50eleqtrrd 2690 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ (𝐶𝑜 suc 𝑦))
52 suceloni 6882 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
53 oecl 7481 . . . . . . . . 9 ((𝐶 ∈ On ∧ suc 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) ∈ On)
5413, 52, 53syl2an 492 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) ∈ On)
55 ontr1 5674 . . . . . . . 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 705 . . . . . 6 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦)))
5857expcom 449 . . . . 5 (𝑦 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦))))
5958adantr 479 . . . 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 bi2.04 374 . . . . . 6 ((𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
6261ralbii 2962 . . . . 5 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ ∀𝑦𝑥 (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
63 r19.21v 2942 . . . . 5 (∀𝑦𝑥 (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
6462, 63bitri 262 . . . 4 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
65 limsuc 6918 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
6665biimpa 499 . . . . . . . . 9 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
67 elex 3184 . . . . . . . . . . . . 13 (suc 𝐴𝑥 → suc 𝐴 ∈ V)
68 sucexb 6878 . . . . . . . . . . . . . 14 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69 sucidg 5706 . . . . . . . . . . . . . 14 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7068, 69sylbir 223 . . . . . . . . . . . . 13 (suc 𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7167, 70syl 17 . . . . . . . . . . . 12 (suc 𝐴𝑥𝐴 ∈ suc 𝐴)
72 eleq2 2676 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → (𝐴𝑦𝐴 ∈ suc 𝐴))
73 oveq2 6535 . . . . . . . . . . . . . . 15 (𝑦 = suc 𝐴 → (𝐶𝑜 𝑦) = (𝐶𝑜 suc 𝐴))
7473eleq2d 2672 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
7572, 74imbi12d 332 . . . . . . . . . . . . 13 (𝑦 = suc 𝐴 → ((𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) ↔ (𝐴 ∈ suc 𝐴 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7675rspcv 3277 . . . . . . . . . . . 12 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐴 ∈ suc 𝐴 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7771, 76mpid 42 . . . . . . . . . . 11 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
7877anc2li 577 . . . . . . . . . 10 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (suc 𝐴𝑥 ∧ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7973eliuni 4456 . . . . . . . . . 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 479 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦)))
8313adantl 480 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → 𝐶 ∈ On)
84 simpl 471 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → Lim 𝑥)
8523adantl 480 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → ∅ ∈ 𝐶)
86 vex 3175 . . . . . . . . . . 11 𝑥 ∈ V
87 oelim 7478 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
8886, 87mpanlr1 717 . . . . . . . . . 10 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
8983, 84, 85, 88syl21anc 1316 . . . . . . . . 9 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
9089adantlr 746 . . . . . . . 8 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
9190eleq2d 2672 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦)))
9282, 91sylibrd 247 . . . . . 6 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)))
9392ex 448 . . . . 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 230 . . 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 6930 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
9796impancom 454 1 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  cdif 3536  c0 3873   ciun 4449  Oncon0 5626  Lim wlim 5627  suc csuc 5628  (class class class)co 6527  1𝑜c1o 7417  2𝑜c2o 7418   ·𝑜 comu 7422  𝑜 coe 7423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-omul 7429  df-oexp 7430
This theorem is referenced by:  oeord  7532  oecan  7533  oeworde  7537  oelimcl  7544
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