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Theorem oeordi 8534
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 7365 . . . . 5 (𝑥 = suc 𝐴 → (𝐶o 𝑥) = (𝐶o suc 𝐴))
21eleq2d 2823 . . . 4 (𝑥 = suc 𝐴 → ((𝐶o 𝐴) ∈ (𝐶o 𝑥) ↔ (𝐶o 𝐴) ∈ (𝐶o suc 𝐴)))
32imbi2d 340 . . 3 (𝑥 = suc 𝐴 → ((𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o suc 𝐴))))
4 oveq2 7365 . . . . 5 (𝑥 = 𝑦 → (𝐶o 𝑥) = (𝐶o 𝑦))
54eleq2d 2823 . . . 4 (𝑥 = 𝑦 → ((𝐶o 𝐴) ∈ (𝐶o 𝑥) ↔ (𝐶o 𝐴) ∈ (𝐶o 𝑦)))
65imbi2d 340 . . 3 (𝑥 = 𝑦 → ((𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
7 oveq2 7365 . . . . 5 (𝑥 = suc 𝑦 → (𝐶o 𝑥) = (𝐶o suc 𝑦))
87eleq2d 2823 . . . 4 (𝑥 = suc 𝑦 → ((𝐶o 𝐴) ∈ (𝐶o 𝑥) ↔ (𝐶o 𝐴) ∈ (𝐶o suc 𝑦)))
98imbi2d 340 . . 3 (𝑥 = suc 𝑦 → ((𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o suc 𝑦))))
10 oveq2 7365 . . . . 5 (𝑥 = 𝐵 → (𝐶o 𝑥) = (𝐶o 𝐵))
1110eleq2d 2823 . . . 4 (𝑥 = 𝐵 → ((𝐶o 𝐴) ∈ (𝐶o 𝑥) ↔ (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
1211imbi2d 340 . . 3 (𝑥 = 𝐵 → ((𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝐵))))
13 eldifi 4086 . . . . . . . 8 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
14 oecl 8483 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
1513, 14sylan 580 . . . . . . 7 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
16 om1 8489 . . . . . . 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 8448 . . . . . . . . 9 (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o𝐶))
1918simprbi 497 . . . . . . . 8 (𝐶 ∈ (On ∖ 2o) → 1o𝐶)
2019adantr 481 . . . . . . 7 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 1o𝐶)
2113adantr 481 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 𝐶 ∈ On)
22 simpr 485 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
23 dif20el 8451 . . . . . . . . . 10 (𝐶 ∈ (On ∖ 2o) → ∅ ∈ 𝐶)
2423adantr 481 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25 oen0 8533 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶o 𝐴))
2621, 22, 24, 25syl21anc 836 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶o 𝐴))
27 omordi 8513 . . . . . . . 8 (((𝐶 ∈ On ∧ (𝐶o 𝐴) ∈ On) ∧ ∅ ∈ (𝐶o 𝐴)) → (1o𝐶 → ((𝐶o 𝐴) ·o 1o) ∈ ((𝐶o 𝐴) ·o 𝐶)))
2821, 15, 26, 27syl21anc 836 . . . . . . 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 2839 . . . . 5 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ ((𝐶o 𝐴) ·o 𝐶))
31 oesuc 8473 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o suc 𝐴) = ((𝐶o 𝐴) ·o 𝐶))
3213, 31sylan 580 . . . . 5 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o suc 𝐴) = ((𝐶o 𝐴) ·o 𝐶))
3330, 32eleqtrrd 2841 . . . 4 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ (𝐶o suc 𝐴))
3433expcom 414 . . 3 (𝐴 ∈ On → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o suc 𝐴)))
35 oecl 8483 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ On)
3613, 35sylan 580 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ On)
37 om1 8489 . . . . . . . . . 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 481 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → 1o𝐶)
4013adantr 481 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
41 simpr 485 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → 𝑦 ∈ On)
4223adantr 481 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ∅ ∈ 𝐶)
43 oen0 8533 . . . . . . . . . . . 12 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶o 𝑦))
4440, 41, 42, 43syl21anc 836 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶o 𝑦))
45 omordi 8513 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝐶o 𝑦) ∈ On) ∧ ∅ ∈ (𝐶o 𝑦)) → (1o𝐶 → ((𝐶o 𝑦) ·o 1o) ∈ ((𝐶o 𝑦) ·o 𝐶)))
4640, 36, 44, 45syl21anc 836 . . . . . . . . . 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 2839 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ ((𝐶o 𝑦) ·o 𝐶))
49 oesuc 8473 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶o suc 𝑦) = ((𝐶o 𝑦) ·o 𝐶))
5013, 49sylan 580 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o suc 𝑦) = ((𝐶o 𝑦) ·o 𝐶))
5148, 50eleqtrrd 2841 . . . . . . 7 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ (𝐶o suc 𝑦))
52 onsuc 7746 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
53 oecl 8483 . . . . . . . . 9 ((𝐶 ∈ On ∧ suc 𝑦 ∈ On) → (𝐶o suc 𝑦) ∈ On)
5413, 52, 53syl2an 596 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o suc 𝑦) ∈ On)
55 ontr1 6363 . . . . . . . 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 692 . . . . . 6 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ((𝐶o 𝐴) ∈ (𝐶o 𝑦) → (𝐶o 𝐴) ∈ (𝐶o suc 𝑦)))
5857expcom 414 . . . . 5 (𝑦 ∈ On → (𝐶 ∈ (On ∖ 2o) → ((𝐶o 𝐴) ∈ (𝐶o 𝑦) → (𝐶o 𝐴) ∈ (𝐶o suc 𝑦))))
5958adantr 481 . . . 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 bi2.04 388 . . . . . 6 ((𝐴𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑦))) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
6261ralbii 3096 . . . . 5 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑦))) ↔ ∀𝑦𝑥 (𝐶 ∈ (On ∖ 2o) → (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
63 r19.21v 3176 . . . . 5 (∀𝑦𝑥 (𝐶 ∈ (On ∖ 2o) → (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))) ↔ (𝐶 ∈ (On ∖ 2o) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
6462, 63bitri 274 . . . 4 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑦))) ↔ (𝐶 ∈ (On ∖ 2o) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
65 limsuc 7785 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
6665biimpa 477 . . . . . . . . 9 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
67 elex 3463 . . . . . . . . . . . . 13 (suc 𝐴𝑥 → suc 𝐴 ∈ V)
68 sucexb 7739 . . . . . . . . . . . . . 14 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69 sucidg 6398 . . . . . . . . . . . . . 14 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7068, 69sylbir 234 . . . . . . . . . . . . 13 (suc 𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7167, 70syl 17 . . . . . . . . . . . 12 (suc 𝐴𝑥𝐴 ∈ suc 𝐴)
72 eleq2 2826 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → (𝐴𝑦𝐴 ∈ suc 𝐴))
73 oveq2 7365 . . . . . . . . . . . . . . 15 (𝑦 = suc 𝐴 → (𝐶o 𝑦) = (𝐶o suc 𝐴))
7473eleq2d 2823 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → ((𝐶o 𝐴) ∈ (𝐶o 𝑦) ↔ (𝐶o 𝐴) ∈ (𝐶o suc 𝐴)))
7572, 74imbi12d 344 . . . . . . . . . . . . 13 (𝑦 = suc 𝐴 → ((𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦)) ↔ (𝐴 ∈ suc 𝐴 → (𝐶o 𝐴) ∈ (𝐶o suc 𝐴))))
7675rspcv 3577 . . . . . . . . . . . 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 556 . . . . . . . . . 10 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦)) → (suc 𝐴𝑥 ∧ (𝐶o 𝐴) ∈ (𝐶o suc 𝐴))))
7973eliuni 4960 . . . . . . . . . 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 481 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦)) → (𝐶o 𝐴) ∈ 𝑦𝑥 (𝐶o 𝑦)))
8313adantl 482 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
84 simpl 483 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → Lim 𝑥)
8523adantl 482 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → ∅ ∈ 𝐶)
86 vex 3449 . . . . . . . . . . 11 𝑥 ∈ V
87 oelim 8480 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
8886, 87mpanlr1 704 . . . . . . . . . 10 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
8983, 84, 85, 88syl21anc 836 . . . . . . . . 9 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
9089adantlr 713 . . . . . . . 8 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
9190eleq2d 2823 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ∈ (𝐶o 𝑥) ↔ (𝐶o 𝐴) ∈ 𝑦𝑥 (𝐶o 𝑦)))
9282, 91sylibrd 258 . . . . . 6 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦)) → (𝐶o 𝐴) ∈ (𝐶o 𝑥)))
9392ex 413 . . . . 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, 94biimtrid 241 . . 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 7798 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
9796impancom 452 1 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cdif 3907  c0 4282   ciun 4954  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7357  1oc1o 8405  2oc2o 8406   ·o comu 8410  o coe 8411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-oexp 8418
This theorem is referenced by:  oeord  8535  oecan  8536  oeworde  8540  oelimcl  8547  oeord2lim  41629  oeord2i  41630  omcl2  41652
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