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Theorem oeordi 8624
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 7439 . . . . 5 (𝑥 = suc 𝐴 → (𝐶o 𝑥) = (𝐶o suc 𝐴))
21eleq2d 2825 . . . 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 7439 . . . . 5 (𝑥 = 𝑦 → (𝐶o 𝑥) = (𝐶o 𝑦))
54eleq2d 2825 . . . 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 7439 . . . . 5 (𝑥 = suc 𝑦 → (𝐶o 𝑥) = (𝐶o suc 𝑦))
87eleq2d 2825 . . . 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 7439 . . . . 5 (𝑥 = 𝐵 → (𝐶o 𝑥) = (𝐶o 𝐵))
1110eleq2d 2825 . . . 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 4141 . . . . . . . 8 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
14 oecl 8574 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
1513, 14sylan 580 . . . . . . 7 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
16 om1 8579 . . . . . . 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 8539 . . . . . . . . 9 (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o𝐶))
1918simprbi 496 . . . . . . . 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 484 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
23 dif20el 8542 . . . . . . . . . 10 (𝐶 ∈ (On ∖ 2o) → ∅ ∈ 𝐶)
2423adantr 480 . . . . . . . . 9 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25 oen0 8623 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶o 𝐴))
2621, 22, 24, 25syl21anc 838 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶o 𝐴))
27 omordi 8603 . . . . . . . 8 (((𝐶 ∈ On ∧ (𝐶o 𝐴) ∈ On) ∧ ∅ ∈ (𝐶o 𝐴)) → (1o𝐶 → ((𝐶o 𝐴) ·o 1o) ∈ ((𝐶o 𝐴) ·o 𝐶)))
2821, 15, 26, 27syl21anc 838 . . . . . . 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 2840 . . . . 5 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ ((𝐶o 𝐴) ·o 𝐶))
31 oesuc 8564 . . . . . 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 2842 . . . 4 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ (𝐶o suc 𝐴))
3433expcom 413 . . 3 (𝐴 ∈ On → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o suc 𝐴)))
35 oecl 8574 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ On)
3613, 35sylan 580 . . . . . . . . . 10 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ On)
37 om1 8579 . . . . . . . . . 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 484 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → 𝑦 ∈ On)
4223adantr 480 . . . . . . . . . . . 12 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ∅ ∈ 𝐶)
43 oen0 8623 . . . . . . . . . . . 12 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶o 𝑦))
4440, 41, 42, 43syl21anc 838 . . . . . . . . . . 11 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶o 𝑦))
45 omordi 8603 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝐶o 𝑦) ∈ On) ∧ ∅ ∈ (𝐶o 𝑦)) → (1o𝐶 → ((𝐶o 𝑦) ·o 1o) ∈ ((𝐶o 𝑦) ·o 𝐶)))
4640, 36, 44, 45syl21anc 838 . . . . . . . . . 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 2840 . . . . . . . 8 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ ((𝐶o 𝑦) ·o 𝐶))
49 oesuc 8564 . . . . . . . . 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 2842 . . . . . . 7 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶o 𝑦) ∈ (𝐶o suc 𝑦))
52 onsuc 7831 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
53 oecl 8574 . . . . . . . . 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 6432 . . . . . . . 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 694 . . . . . 6 ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ((𝐶o 𝐴) ∈ (𝐶o 𝑦) → (𝐶o 𝐴) ∈ (𝐶o suc 𝑦)))
5857expcom 413 . . . . 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 bi2.04 387 . . . . . 6 ((𝐴𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑦))) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
6261ralbii 3091 . . . . 5 (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝑦))) ↔ ∀𝑦𝑥 (𝐶 ∈ (On ∖ 2o) → (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦))))
63 r19.21v 3178 . . . . 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 7870 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
6665biimpa 476 . . . . . . . . 9 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
67 elex 3499 . . . . . . . . . . . . 13 (suc 𝐴𝑥 → suc 𝐴 ∈ V)
68 sucexb 7824 . . . . . . . . . . . . . 14 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69 sucidg 6467 . . . . . . . . . . . . . 14 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7068, 69sylbir 235 . . . . . . . . . . . . 13 (suc 𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7167, 70syl 17 . . . . . . . . . . . 12 (suc 𝐴𝑥𝐴 ∈ suc 𝐴)
72 eleq2 2828 . . . . . . . . . . . . . 14 (𝑦 = suc 𝐴 → (𝐴𝑦𝐴 ∈ suc 𝐴))
73 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑦 = suc 𝐴 → (𝐶o 𝑦) = (𝐶o suc 𝐴))
7473eleq2d 2825 . . . . . . . . . . . . . 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 3618 . . . . . . . . . . . 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 555 . . . . . . . . . 10 (suc 𝐴𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (𝐶o 𝐴) ∈ (𝐶o 𝑦)) → (suc 𝐴𝑥 ∧ (𝐶o 𝐴) ∈ (𝐶o suc 𝐴))))
7973eliuni 5002 . . . . . . . . . 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 482 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → Lim 𝑥)
8523adantl 481 . . . . . . . . . 10 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → ∅ ∈ 𝐶)
86 vex 3482 . . . . . . . . . . 11 𝑥 ∈ V
87 oelim 8571 . . . . . . . . . . 11 (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
8886, 87mpanlr1 706 . . . . . . . . . 10 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
8983, 84, 85, 88syl21anc 838 . . . . . . . . 9 ((Lim 𝑥𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
9089adantlr 715 . . . . . . . 8 (((Lim 𝑥𝐴𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝑥) = 𝑦𝑥 (𝐶o 𝑦))
9190eleq2d 2825 . . . . . . 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 412 . . . . 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 242 . . 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 7883 . 2 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐶 ∈ (On ∖ 2o) → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
9796impancom 451 1 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cdif 3960  c0 4339   ciun 4996  Oncon0 6386  Lim wlim 6387  suc csuc 6388  (class class class)co 7431  1oc1o 8498  2oc2o 8499   ·o comu 8503  o coe 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511
This theorem is referenced by:  oeord  8625  oecan  8626  oeworde  8630  oelimcl  8637  oeord2lim  43299  oeord2i  43300  omcl2  43323
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