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Theorem oeordi 7712
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 6698 . . . . 5 (𝑥 = suc 𝐴 → (𝐶𝑜 𝑥) = (𝐶𝑜 suc 𝐴))
21eleq2d 2716 . . . 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 6698 . . . . 5 (𝑥 = 𝑦 → (𝐶𝑜 𝑥) = (𝐶𝑜 𝑦))
54eleq2d 2716 . . . 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 6698 . . . . 5 (𝑥 = suc 𝑦 → (𝐶𝑜 𝑥) = (𝐶𝑜 suc 𝑦))
87eleq2d 2716 . . . 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 6698 . . . . 5 (𝑥 = 𝐵 → (𝐶𝑜 𝑥) = (𝐶𝑜 𝐵))
1110eleq2d 2716 . . . 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 3765 . . . . . . . 8 (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
14 oecl 7662 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
1513, 14sylan 487 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
16 om1 7667 . . . . . . 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 7627 . . . . . . . . 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 7630 . . . . . . . . . 10 (𝐶 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐶)
2423adantr 480 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25 oen0 7711 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶𝑜 𝐴))
2621, 22, 24, 25syl21anc 1365 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶𝑜 𝐴))
27 omordi 7691 . . . . . . . 8 (((𝐶 ∈ On ∧ (𝐶𝑜 𝐴) ∈ On) ∧ ∅ ∈ (𝐶𝑜 𝐴)) → (1𝑜𝐶 → ((𝐶𝑜 𝐴) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶)))
2821, 15, 26, 27syl21anc 1365 . . . . . . 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 2731 . . . . 5 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶))
31 oesuc 7652 . . . . . 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 2733 . . . 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 7662 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ On)
3613, 35sylan 487 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ On)
37 om1 7667 . . . . . . . . . 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 7711 . . . . . . . . . . . 12 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶𝑜 𝑦))
4440, 41, 42, 43syl21anc 1365 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶𝑜 𝑦))
45 omordi 7691 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝐶𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐶𝑜 𝑦)) → (1𝑜𝐶 → ((𝐶𝑜 𝑦) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶)))
4640, 36, 44, 45syl21anc 1365 . . . . . . . . . 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 2731 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶))
49 oesuc 7652 . . . . . . . . 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 2733 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ (𝐶𝑜 suc 𝑦))
52 suceloni 7055 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
53 oecl 7662 . . . . . . . . 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 5809 . . . . . . . 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 710 . . . . . 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 3009 . . . . 5 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ ∀𝑦𝑥 (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
63 r19.21v 2989 . . . . 5 (∀𝑦𝑥 (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
6462, 63bitri 264 . . . 4 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
65 limsuc 7091 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
6665biimpa 500 . . . . . . . . 9 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
67 elex 3243 . . . . . . . . . . . . 13 (suc 𝐴𝑥 → suc 𝐴 ∈ V)
68 sucexb 7051 . . . . . . . . . . . . . 14 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69 sucidg 5841 . . . . . . . . . . . . . 14 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7068, 69sylbir 225 . . . . . . . . . . . . 13 (suc 𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7167, 70syl 17 . . . . . . . . . . . 12 (suc 𝐴𝑥𝐴 ∈ suc 𝐴)
72 eleq2 2719 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → (𝐴𝑦𝐴 ∈ suc 𝐴))
73 oveq2 6698 . . . . . . . . . . . . . . 15 (𝑦 = suc 𝐴 → (𝐶𝑜 𝑦) = (𝐶𝑜 suc 𝐴))
7473eleq2d 2716 . . . . . . . . . . . . . 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 3336 . . . . . . . . . . . 12 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐴 ∈ suc 𝐴 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7771, 76mpid 44 . . . . . . . . . . 11 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
7877anc2li 579 . . . . . . . . . 10 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (suc 𝐴𝑥 ∧ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7973eliuni 4558 . . . . . . . . . 10 ((suc 𝐴𝑥 ∧ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)) → (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦))
8078, 79syl6 35 . . . . . . . . 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 3234 . . . . . . . . . . 11 𝑥 ∈ V
87 oelim 7659 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
8886, 87mpanlr1 722 . . . . . . . . . 10 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
8983, 84, 85, 88syl21anc 1365 . . . . . . . . 9 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
9089adantlr 751 . . . . . . . 8 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
9190eleq2d 2716 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦)))
9282, 91sylibrd 249 . . . . . 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 232 . . 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 7103 . 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 1523  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  c0 3948   ciun 4552  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690  1𝑜c1o 7598  2𝑜c2o 7599   ·𝑜 comu 7603  𝑜 coe 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611
This theorem is referenced by:  oeord  7713  oecan  7714  oeworde  7718  oelimcl  7725
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