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Mirrors  >  Home  >  MPE Home  >  Th. List  >  oeordi Structured version   Visualization version   GIF version

Theorem oeordi 7821
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 6801 . . . . 5 (𝑥 = suc 𝐴 → (𝐶𝑜 𝑥) = (𝐶𝑜 suc 𝐴))
21eleq2d 2836 . . . 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 6801 . . . . 5 (𝑥 = 𝑦 → (𝐶𝑜 𝑥) = (𝐶𝑜 𝑦))
54eleq2d 2836 . . . 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 6801 . . . . 5 (𝑥 = suc 𝑦 → (𝐶𝑜 𝑥) = (𝐶𝑜 suc 𝑦))
87eleq2d 2836 . . . 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 6801 . . . . 5 (𝑥 = 𝐵 → (𝐶𝑜 𝑥) = (𝐶𝑜 𝐵))
1110eleq2d 2836 . . . 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 3883 . . . . . . . 8 (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
14 oecl 7771 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
1513, 14sylan 569 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
16 om1 7776 . . . . . . 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 7736 . . . . . . . . 9 (𝐶 ∈ (On ∖ 2𝑜) ↔ (𝐶 ∈ On ∧ 1𝑜𝐶))
1918simprbi 484 . . . . . . . 8 (𝐶 ∈ (On ∖ 2𝑜) → 1𝑜𝐶)
2019adantr 466 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 1𝑜𝐶)
2113adantr 466 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐶 ∈ On)
22 simpr 471 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
23 dif20el 7739 . . . . . . . . . 10 (𝐶 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐶)
2423adantr 466 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25 oen0 7820 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶𝑜 𝐴))
2621, 22, 24, 25syl21anc 1475 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶𝑜 𝐴))
27 omordi 7800 . . . . . . . 8 (((𝐶 ∈ On ∧ (𝐶𝑜 𝐴) ∈ On) ∧ ∅ ∈ (𝐶𝑜 𝐴)) → (1𝑜𝐶 → ((𝐶𝑜 𝐴) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶)))
2821, 15, 26, 27syl21anc 1475 . . . . . . 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 2851 . . . . 5 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ ((𝐶𝑜 𝐴) ·𝑜 𝐶))
31 oesuc 7761 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 suc 𝐴) = ((𝐶𝑜 𝐴) ·𝑜 𝐶))
3213, 31sylan 569 . . . . 5 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 suc 𝐴) = ((𝐶𝑜 𝐴) ·𝑜 𝐶))
3330, 32eleqtrrd 2853 . . . 4 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))
3433expcom 398 . . 3 (𝐴 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴)))
35 oecl 7771 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ On)
3613, 35sylan 569 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ On)
37 om1 7776 . . . . . . . . . 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 466 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 1𝑜𝐶)
4013adantr 466 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
41 simpr 471 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝑦 ∈ On)
4223adantr 466 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ 𝐶)
43 oen0 7820 . . . . . . . . . . . 12 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶𝑜 𝑦))
4440, 41, 42, 43syl21anc 1475 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶𝑜 𝑦))
45 omordi 7800 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝐶𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐶𝑜 𝑦)) → (1𝑜𝐶 → ((𝐶𝑜 𝑦) ·𝑜 1𝑜) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶)))
4640, 36, 44, 45syl21anc 1475 . . . . . . . . . 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 2851 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ ((𝐶𝑜 𝑦) ·𝑜 𝐶))
49 oesuc 7761 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) = ((𝐶𝑜 𝑦) ·𝑜 𝐶))
5013, 49sylan 569 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) = ((𝐶𝑜 𝑦) ·𝑜 𝐶))
5148, 50eleqtrrd 2853 . . . . . . 7 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 𝑦) ∈ (𝐶𝑜 suc 𝑦))
52 suceloni 7160 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
53 oecl 7771 . . . . . . . . 9 ((𝐶 ∈ On ∧ suc 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) ∈ On)
5413, 52, 53syl2an 583 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶𝑜 suc 𝑦) ∈ On)
55 ontr1 5914 . . . . . . . 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 674 . . . . . 6 ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦)))
5857expcom 398 . . . . 5 (𝑦 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝑦))))
5958adantr 466 . . . 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 3129 . . . . 5 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))) ↔ ∀𝑦𝑥 (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦))))
63 r19.21v 3109 . . . . 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 7196 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
6665biimpa 462 . . . . . . . . 9 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
67 elex 3364 . . . . . . . . . . . . 13 (suc 𝐴𝑥 → suc 𝐴 ∈ V)
68 sucexb 7156 . . . . . . . . . . . . . 14 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69 sucidg 5946 . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → (𝐴𝑦𝐴 ∈ suc 𝐴))
73 oveq2 6801 . . . . . . . . . . . . . . 15 (𝑦 = suc 𝐴 → (𝐶𝑜 𝑦) = (𝐶𝑜 suc 𝐴))
7473eleq2d 2836 . . . . . . . . . . . . . 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 3456 . . . . . . . . . . . 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 545 . . . . . . . . . 10 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (suc 𝐴𝑥 ∧ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 suc 𝐴))))
7973eliuni 4660 . . . . . . . . . 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 466 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝑦)) → (𝐶𝑜 𝐴) ∈ 𝑦𝑥 (𝐶𝑜 𝑦)))
8313adantl 467 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → 𝐶 ∈ On)
84 simpl 468 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → Lim 𝑥)
8523adantl 467 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → ∅ ∈ 𝐶)
86 vex 3354 . . . . . . . . . . 11 𝑥 ∈ V
87 oelim 7768 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
8886, 87mpanlr1 686 . . . . . . . . . 10 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
8983, 84, 85, 88syl21anc 1475 . . . . . . . . 9 ((Lim 𝑥𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
9089adantlr 694 . . . . . . . 8 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝑥) = 𝑦𝑥 (𝐶𝑜 𝑦))
9190eleq2d 2836 . . . . . . 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 397 . . . . 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 7208 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
9796impancom 439 1 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cdif 3720  c0 4063   ciun 4654  Oncon0 5866  Lim wlim 5867  suc csuc 5868  (class class class)co 6793  1𝑜c1o 7706  2𝑜c2o 7707   ·𝑜 comu 7711  𝑜 coe 7712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-oexp 7719
This theorem is referenced by:  oeord  7822  oecan  7823  oeworde  7827  oelimcl  7834
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