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Theorem omordi 8382
Description: Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)
Assertion
Ref Expression
omordi (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Proof of Theorem omordi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onelon 6290 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 413 . . . . 5 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
3 eleq2 2829 . . . . . . . . . 10 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
4 oveq2 7279 . . . . . . . . . . 11 (𝑥 = ∅ → (𝐶 ·o 𝑥) = (𝐶 ·o ∅))
54eleq2d 2826 . . . . . . . . . 10 (𝑥 = ∅ → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
63, 5imbi12d 345 . . . . . . . . 9 (𝑥 = ∅ → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))))
7 eleq2 2829 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
8 oveq2 7279 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝑦))
98eleq2d 2826 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))
107, 9imbi12d 345 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))))
11 eleq2 2829 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
12 oveq2 7279 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o suc 𝑦))
1312eleq2d 2826 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
1411, 13imbi12d 345 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))
15 eleq2 2829 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
16 oveq2 7279 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝐵))
1716eleq2d 2826 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
1815, 17imbi12d 345 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
19 noel 4270 . . . . . . . . . . 11 ¬ 𝐴 ∈ ∅
2019pm2.21i 119 . . . . . . . . . 10 (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))
2120a1i 11 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
22 elsuci 6331 . . . . . . . . . . . . . . 15 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
23 omcl 8351 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·o 𝑦) ∈ On)
24 simpl 483 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
2523, 24jca 512 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On))
26 oaword1 8368 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → (𝐶 ·o 𝑦) ⊆ ((𝐶 ·o 𝑦) +o 𝐶))
2726sseld 3925 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
2827imim2d 57 . . . . . . . . . . . . . . . . . . 19 (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶))))
2928imp 407 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))) → (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3029adantrl 713 . . . . . . . . . . . . . . . . 17 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
31 oaord1 8367 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3231biimpa 477 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
33 oveq2 7279 . . . . . . . . . . . . . . . . . . . 20 (𝐴 = 𝑦 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝑦))
3433eleq1d 2825 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑦 → ((𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶) ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3532, 34syl5ibrcom 246 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3635adantrr 714 . . . . . . . . . . . . . . . . 17 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3730, 36jaod 856 . . . . . . . . . . . . . . . 16 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐴𝑦𝐴 = 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3825, 37sylan 580 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐴𝑦𝐴 = 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3922, 38syl5 34 . . . . . . . . . . . . . 14 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
40 omsuc 8341 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·o suc 𝑦) = ((𝐶 ·o 𝑦) +o 𝐶))
4140eleq2d 2826 . . . . . . . . . . . . . . 15 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
4241adantr 481 . . . . . . . . . . . . . 14 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
4339, 42sylibrd 258 . . . . . . . . . . . . 13 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
4443exp43 437 . . . . . . . . . . . 12 (𝐶 ∈ On → (𝑦 ∈ On → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
4544com12 32 . . . . . . . . . . 11 (𝑦 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
4645adantld 491 . . . . . . . . . 10 (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
4746impd 411 . . . . . . . . 9 (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))))
48 id 22 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ∈ On ∧ Lim 𝑥))
4948ad2ant2r 744 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ∈ On ∧ Lim 𝑥))
50 limsuc 7690 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
5150biimpa 477 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
52 oveq2 7279 . . . . . . . . . . . . . . . . . . 19 (𝑦 = suc 𝐴 → (𝐶 ·o 𝑦) = (𝐶 ·o suc 𝐴))
5352ssiun2s 4983 . . . . . . . . . . . . . . . . . 18 (suc 𝐴𝑥 → (𝐶 ·o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 ·o 𝑦))
5451, 53syl 17 . . . . . . . . . . . . . . . . 17 ((Lim 𝑥𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 ·o 𝑦))
5554adantll 711 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 ·o 𝑦))
56 vex 3435 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
57 omlim 8348 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 ·o 𝑥) = 𝑦𝑥 (𝐶 ·o 𝑦))
5856, 57mpanr1 700 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ·o 𝑥) = 𝑦𝑥 (𝐶 ·o 𝑦))
5958adantr 481 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·o 𝑥) = 𝑦𝑥 (𝐶 ·o 𝑦))
6055, 59sseqtrrd 3967 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ (𝐶 ·o 𝑥))
6149, 60sylan 580 . . . . . . . . . . . . . 14 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ (𝐶 ·o 𝑥))
62 omcl 8351 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o 𝐴) ∈ On)
63 oaord1 8367 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·o 𝐴) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
6462, 63sylan 580 . . . . . . . . . . . . . . . . . . 19 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
6564anabss1 663 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
6665biimpa 477 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶))
67 omsuc 8341 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o suc 𝐴) = ((𝐶 ·o 𝐴) +o 𝐶))
6867adantr 481 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o suc 𝐴) = ((𝐶 ·o 𝐴) +o 𝐶))
6966, 68eleqtrrd 2844 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
7069adantrl 713 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
7170adantr 481 . . . . . . . . . . . . . 14 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
7261, 71sseldd 3927 . . . . . . . . . . . . 13 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))
7372exp53 448 . . . . . . . . . . . 12 (𝐶 ∈ On → (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐶 → (𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))))
7473com13 88 . . . . . . . . . . 11 (Lim 𝑥 → (𝐴 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))))
7574imp4c 424 . . . . . . . . . 10 (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))
7675a1dd 50 . . . . . . . . 9 (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)))))
776, 10, 14, 18, 21, 47, 76tfinds3 7705 . . . . . . . 8 (𝐵 ∈ On → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
7877com23 86 . . . . . . 7 (𝐵 ∈ On → (𝐴𝐵 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
7978exp4a 432 . . . . . 6 (𝐵 ∈ On → (𝐴𝐵 → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
8079exp4a 432 . . . . 5 (𝐵 ∈ On → (𝐴𝐵 → (𝐴 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))))
812, 80mpdd 43 . . . 4 (𝐵 ∈ On → (𝐴𝐵 → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
8281com34 91 . . 3 (𝐵 ∈ On → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ On → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
8382com24 95 . 2 (𝐵 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
8483imp31 418 1 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844   = wceq 1542  wcel 2110  wral 3066  Vcvv 3431  wss 3892  c0 4262   ciun 4930  Oncon0 6265  Lim wlim 6266  suc csuc 6267  (class class class)co 7271   +o coa 8285   ·o comu 8286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-oadd 8292  df-omul 8293
This theorem is referenced by:  omord2  8383  omcan  8385  odi  8395  omass  8396  oen0  8402  oeordi  8403  oeordsuc  8410
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