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Theorem oeordi 8584
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 7414 . . . . 5 (𝑥 = suc 𝐴 → (𝐶o 𝑥) = (𝐶o suc 𝐴))
21eleq2d 2820 . . . 4 (𝑥 = suc 𝐴 → ((𝐶o 𝐴) ∈ (𝐶o 𝑥) ↔ (𝐶o 𝐴) ∈ (𝐶o suc 𝐴)))
32imbi2d 341 . . 3 (𝑥 = suc 𝐴 → ((𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o suc 𝐴))))
4 oveq2 7414 . . . . 5 (𝑥 = 𝑦 → (𝐶o 𝑥) = (𝐶o 𝑦))
54eleq2d 2820 . . . 4 (𝑥 = 𝑦 → ((𝐶o 𝐴) ∈ (𝐶o 𝑥) ↔ (𝐶o 𝐴) ∈ (𝐶o 𝑦)))
65imbi2d 341 . . 3 (𝑥 = 𝑦 → ((𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
7 oveq2 7414 . . . . 5 (𝑥 = suc 𝑦 → (𝐶o 𝑥) = (𝐶o suc 𝑦))
87eleq2d 2820 . . . 4 (𝑥 = suc 𝑦 → ((𝐶o 𝐴) ∈ (𝐶o 𝑥) ↔ (𝐶o 𝐴) ∈ (𝐶o suc 𝑦)))
98imbi2d 341 . . 3 (𝑥 = suc 𝑦 → ((𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o suc 𝑦))))
10 oveq2 7414 . . . . 5 (𝑥 = 𝐵 → (𝐶o 𝑥) = (𝐶o 𝐵))
1110eleq2d 2820 . . . 4 (𝑥 = 𝐵 → ((𝐶o 𝐴) ∈ (𝐶o 𝑥) ↔ (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
1211imbi2d 341 . . 3 (𝑥 = 𝐵 → ((𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝐵))))
13 eldifi 4126 . . . . . . . 8 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
14 oecl 8534 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
1513, 14sylan 581 . . . . . . 7 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
16 om1 8539 . . . . . . 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 8499 . . . . . . . . 9 (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o𝐶))
1918simprbi 498 . . . . . . . 8 (𝐶 ∈ (On ∖ 2o) → 1o𝐶)
2019adantr 482 . . . . . . 7 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 1o𝐶)
2113adantr 482 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 𝐶 ∈ On)
22 simpr 486 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
23 dif20el 8502 . . . . . . . . . 10 (𝐶 ∈ (On ∖ 2o) → ∅ ∈ 𝐶)
2423adantr 482 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25 oen0 8583 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶o 𝐴))
2621, 22, 24, 25syl21anc 837 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶o 𝐴))
27 omordi 8563 . . . . . . . 8 (((𝐶 ∈ On ∧ (𝐶o 𝐴) ∈ On) ∧ ∅ ∈ (𝐶o 𝐴)) → (1o𝐶 → ((𝐶o 𝐴) ·o 1o) ∈ ((𝐶o 𝐴) ·o 𝐶)))
2821, 15, 26, 27syl21anc 837 . . . . . . 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 2835 . . . . 5 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ ((𝐶o 𝐴) ·o 𝐶))
31 oesuc 8524 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o suc 𝐴) = ((𝐶o 𝐴) ·o 𝐶))
3213, 31sylan 581 . . . . 5 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o suc 𝐴) = ((𝐶o 𝐴) ·o 𝐶))
3330, 32eleqtrrd 2837 . . . 4 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ (𝐶o suc 𝐴))
3433expcom 415 . . 3 (𝐴 ∈ On → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o suc 𝐴)))
35 oecl 8534 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ On)
3613, 35sylan 581 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ On)
37 om1 8539 . . . . . . . . . 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 482 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → 1o𝐶)
4013adantr 482 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
41 simpr 486 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → 𝑦 ∈ On)
4223adantr 482 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ∅ ∈ 𝐶)
43 oen0 8583 . . . . . . . . . . . 12 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶o 𝑦))
4440, 41, 42, 43syl21anc 837 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶o 𝑦))
45 omordi 8563 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝐶o 𝑦) ∈ On) ∧ ∅ ∈ (𝐶o 𝑦)) → (1o𝐶 → ((𝐶o 𝑦) ·o 1o) ∈ ((𝐶o 𝑦) ·o 𝐶)))
4640, 36, 44, 45syl21anc 837 . . . . . . . . . 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 2835 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ ((𝐶o 𝑦) ·o 𝐶))
49 oesuc 8524 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶o suc 𝑦) = ((𝐶o 𝑦) ·o 𝐶))
5013, 49sylan 581 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o suc 𝑦) = ((𝐶o 𝑦) ·o 𝐶))
5148, 50eleqtrrd 2837 . . . . . . 7 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ (𝐶o suc 𝑦))
52 onsuc 7796 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
53 oecl 8534 . . . . . . . . 9 ((𝐶 ∈ On ∧ suc 𝑦 ∈ On) → (𝐶o suc 𝑦) ∈ On)
5413, 52, 53syl2an 597 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o suc 𝑦) ∈ On)
55 ontr1 6408 . . . . . . . 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 693 . . . . . 6 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ((𝐶o 𝐴) ∈ (𝐶o 𝑦) → (𝐶o 𝐴) ∈ (𝐶o suc 𝑦)))
5857expcom 415 . . . . 5 (𝑦 ∈ On → (𝐶 ∈ (On ∖ 2o) → ((𝐶o 𝐴) ∈ (𝐶o 𝑦) → (𝐶o 𝐴) ∈ (𝐶o suc 𝑦))))
5958adantr 482 . . . 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 bi2.04 389 . . . . . 6 ((𝐴𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑦))) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
6261ralbii 3094 . . . . 5 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑦))) ↔ ∀𝑦𝑥 (𝐶 ∈ (On ∖ 2o) → (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
63 r19.21v 3180 . . . . 5 (∀𝑦𝑥 (𝐶 ∈ (On ∖ 2o) → (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))) ↔ (𝐶 ∈ (On ∖ 2o) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
6462, 63bitri 275 . . . 4 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑦))) ↔ (𝐶 ∈ (On ∖ 2o) → ∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
65 limsuc 7835 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
6665biimpa 478 . . . . . . . . 9 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
67 elex 3493 . . . . . . . . . . . . 13 (suc 𝐴𝑥 → suc 𝐴 ∈ V)
68 sucexb 7789 . . . . . . . . . . . . . 14 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69 sucidg 6443 . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → (𝐴𝑦𝐴 ∈ suc 𝐴))
73 oveq2 7414 . . . . . . . . . . . . . . 15 (𝑦 = suc 𝐴 → (𝐶o 𝑦) = (𝐶o suc 𝐴))
7473eleq2d 2820 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → ((𝐶o 𝐴) ∈ (𝐶o 𝑦) ↔ (𝐶o 𝐴) ∈ (𝐶o suc 𝐴)))
7572, 74imbi12d 345 . . . . . . . . . . . . 13 (𝑦 = suc 𝐴 → ((𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦)) ↔ (𝐴 ∈ suc 𝐴 → (𝐶o 𝐴) ∈ (𝐶o suc 𝐴))))
7675rspcv 3609 . . . . . . . . . . . 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 557 . . . . . . . . . 10 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦)) → (suc 𝐴𝑥 ∧ (𝐶o 𝐴) ∈ (𝐶o suc 𝐴))))
7973eliuni 5003 . . . . . . . . . 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 482 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦)) → (𝐶o 𝐴) ∈ 𝑦𝑥 (𝐶o 𝑦)))
8313adantl 483 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
84 simpl 484 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → Lim 𝑥)
8523adantl 483 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → ∅ ∈ 𝐶)
86 vex 3479 . . . . . . . . . . 11 𝑥 ∈ V
87 oelim 8531 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
8886, 87mpanlr1 705 . . . . . . . . . 10 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
8983, 84, 85, 88syl21anc 837 . . . . . . . . 9 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
9089adantlr 714 . . . . . . . 8 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
9190eleq2d 2820 . . . . . . 7 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ∈ (𝐶o 𝑥) ↔ (𝐶o 𝐴) ∈ 𝑦𝑥 (𝐶o 𝑦)))
9282, 91sylibrd 259 . . . . . 6 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦)) → (𝐶o 𝐴) ∈ (𝐶o 𝑥)))
9392ex 414 . . . . 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 7848 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
9796impancom 453 1 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cdif 3945  c0 4322   ciun 4997  Oncon0 6362  Lim wlim 6363  suc csuc 6364  (class class class)co 7406  1oc1o 8456  2oc2o 8457   ·o comu 8461  o coe 8462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-2o 8464  df-oadd 8467  df-omul 8468  df-oexp 8469
This theorem is referenced by:  oeord  8585  oecan  8586  oeworde  8590  oelimcl  8597  oeord2lim  42045  oeord2i  42046  omcl2  42069
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