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Theorem oeordi 7874
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 6852 . . . . 5 (𝑥 = suc 𝐴 → (𝐶𝑜 𝑥) = (𝐶𝑜 suc 𝐴))
21eleq2d 2830 . . . 4 (𝑥 = suc 𝐴 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
32imbi2d 331 . . 3 (𝑥 = suc 𝐴 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
4 oveq2 6852 . . . . 5 (𝑥 = 𝑦 → (𝐶𝑜 𝑥) = (𝐶𝑜 𝑦))
54eleq2d 2830 . . . 4 (𝑥 = 𝑦 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)))
65imbi2d 331 . . 3 (𝑥 = 𝑦 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
7 oveq2 6852 . . . . 5 (𝑥 = suc 𝑦 → (𝐶𝑜 𝑥) = (𝐶𝑜 suc 𝑦))
87eleq2d 2830 . . . 4 (𝑥 = suc 𝑦 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦)))
98imbi2d 331 . . 3 (𝑥 = suc 𝑦 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦))))
10 oveq2 6852 . . . . 5 (𝑥 = 𝐵 → (𝐶𝑜 𝑥) = (𝐶𝑜 𝐵))
1110eleq2d 2830 . . . 4 (𝑥 = 𝐵 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
1211imbi2d 331 . . 3 (𝑥 = 𝐵 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵))))
13 eldifi 3896 . . . . . . . 8 (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
14 oecl 7824 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
1513, 14sylan 575 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
16 om1 7829 . . . . . . 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 7789 . . . . . . . . 9 (𝐶 ∈ (On ∖ 2𝑜) ↔ (𝐶 ∈ On ∧ 1𝑜𝐶))
1918simprbi 490 . . . . . . . 8 (𝐶 ∈ (On ∖ 2𝑜) → 1𝑜𝐶)
2019adantr 472 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 1𝑜𝐶)
2113adantr 472 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐶 ∈ On)
22 simpr 477 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
23 dif20el 7792 . . . . . . . . . 10 (𝐶 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐶)
2423adantr 472 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25 oen0 7873 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶𝑜 𝐴))
2621, 22, 24, 25syl21anc 866 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶𝑜 𝐴))
27 omordi 7853 . . . . . . . 8 (((𝐶 ∈ On ∧ (𝐶𝑜 𝐴) ∈ On) ∧ ∅ ∈ (𝐶𝑜 𝐴)) → (1𝑜𝐶 → ((𝐶𝑜 𝐴) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶)))
2821, 15, 26, 27syl21anc 866 . . . . . . 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 2845 . . . . 5 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶))
31 oesuc 7814 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 suc 𝐴) = ((𝐶𝑜 𝐴) ·𝑜 𝐶))
3213, 31sylan 575 . . . . 5 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 suc 𝐴) = ((𝐶𝑜 𝐴) ·𝑜 𝐶))
3330, 32eleqtrrd 2847 . . . 4 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))
3433expcom 402 . . 3 (𝐴 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
35 oecl 7824 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ On)
3613, 35sylan 575 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ On)
37 om1 7829 . . . . . . . . . 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 472 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 1𝑜𝐶)
4013adantr 472 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
41 simpr 477 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝑦 ∈ On)
4223adantr 472 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ 𝐶)
43 oen0 7873 . . . . . . . . . . . 12 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶𝑜 𝑦))
4440, 41, 42, 43syl21anc 866 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶𝑜 𝑦))
45 omordi 7853 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝐶𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐶𝑜 𝑦)) → (1𝑜𝐶 → ((𝐶𝑜 𝑦) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶)))
4640, 36, 44, 45syl21anc 866 . . . . . . . . . 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 2845 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶))
49 oesuc 7814 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) = ((𝐶𝑜 𝑦) ·𝑜 𝐶))
5013, 49sylan 575 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) = ((𝐶𝑜 𝑦) ·𝑜 𝐶))
5148, 50eleqtrrd 2847 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ (𝐶𝑜 suc 𝑦))
52 suceloni 7213 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
53 oecl 7824 . . . . . . . . 9 ((𝐶 ∈ On ∧ suc 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) ∈ On)
5413, 52, 53syl2an 589 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) ∈ On)
55 ontr1 5956 . . . . . . . 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 685 . . . . . 6 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦)))
5857expcom 402 . . . . 5 (𝑦 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦))))
5958adantr 472 . . . 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 bi2.04 377 . . . . . 6 ((𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
6261ralbii 3127 . . . . 5 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ ∀𝑦𝑥 (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
63 r19.21v 3107 . . . . 5 (∀𝑦𝑥 (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
6462, 63bitri 266 . . . 4 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ (𝐶 ∈ (On ∖ 2𝑜) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
65 limsuc 7249 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
6665biimpa 468 . . . . . . . . 9 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
67 elex 3365 . . . . . . . . . . . . 13 (suc 𝐴𝑥 → suc 𝐴 ∈ V)
68 sucexb 7209 . . . . . . . . . . . . . 14 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69 sucidg 5988 . . . . . . . . . . . . . 14 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7068, 69sylbir 226 . . . . . . . . . . . . 13 (suc 𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7167, 70syl 17 . . . . . . . . . . . 12 (suc 𝐴𝑥𝐴 ∈ suc 𝐴)
72 eleq2 2833 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → (𝐴𝑦𝐴 ∈ suc 𝐴))
73 oveq2 6852 . . . . . . . . . . . . . . 15 (𝑦 = suc 𝐴 → (𝐶𝑜 𝑦) = (𝐶𝑜 suc 𝐴))
7473eleq2d 2830 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
7572, 74imbi12d 335 . . . . . . . . . . . . 13 (𝑦 = suc 𝐴 → ((𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) ↔ (𝐴 ∈ suc 𝐴 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7675rspcv 3458 . . . . . . . . . . . 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 551 . . . . . . . . . 10 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (suc 𝐴𝑥 ∧ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7973eliuni 4684 . . . . . . . . . 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 472 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦)))
8313adantl 473 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → 𝐶 ∈ On)
84 simpl 474 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → Lim 𝑥)
8523adantl 473 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → ∅ ∈ 𝐶)
86 vex 3353 . . . . . . . . . . 11 𝑥 ∈ V
87 oelim 7821 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
8886, 87mpanlr1 697 . . . . . . . . . 10 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
8983, 84, 85, 88syl21anc 866 . . . . . . . . 9 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
9089adantlr 706 . . . . . . . 8 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
9190eleq2d 2830 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥) ↔ (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦)))
9282, 91sylibrd 250 . . . . . 6 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑥)))
9392ex 401 . . . . 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 233 . . 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 7261 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
9796impancom 443 1 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cdif 3731  c0 4081   ciun 4678  Oncon0 5910  Lim wlim 5911  suc csuc 5912  (class class class)co 6844  1𝑜c1o 7759  2𝑜c2o 7760   ·𝑜 comu 7764  𝑜 coe 7765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-2o 7767  df-oadd 7770  df-omul 7771  df-oexp 7772
This theorem is referenced by:  oeord  7875  oecan  7876  oeworde  7880  oelimcl  7887
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