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Theorem oewordri 7717
Description: Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)
Assertion
Ref Expression
oewordri ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))

Proof of Theorem oewordri
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6698 . . . . 5 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
2 oveq2 6698 . . . . 5 (𝑥 = ∅ → (𝐵𝑜 𝑥) = (𝐵𝑜 ∅))
31, 2sseq12d 3667 . . . 4 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 ∅) ⊆ (𝐵𝑜 ∅)))
4 oveq2 6698 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
5 oveq2 6698 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑜 𝑥) = (𝐵𝑜 𝑦))
64, 5sseq12d 3667 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦)))
7 oveq2 6698 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
8 oveq2 6698 . . . . 5 (𝑥 = suc 𝑦 → (𝐵𝑜 𝑥) = (𝐵𝑜 suc 𝑦))
97, 8sseq12d 3667 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 suc 𝑦) ⊆ (𝐵𝑜 suc 𝑦)))
10 oveq2 6698 . . . . 5 (𝑥 = 𝐶 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐶))
11 oveq2 6698 . . . . 5 (𝑥 = 𝐶 → (𝐵𝑜 𝑥) = (𝐵𝑜 𝐶))
1210, 11sseq12d 3667 . . . 4 (𝑥 = 𝐶 → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
13 onelon 5786 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
14 oe0 7647 . . . . . . 7 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
1513, 14syl 17 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 ∅) = 1𝑜)
16 oe0 7647 . . . . . . 7 (𝐵 ∈ On → (𝐵𝑜 ∅) = 1𝑜)
1716adantr 480 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐵𝑜 ∅) = 1𝑜)
1815, 17eqtr4d 2688 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 ∅) = (𝐵𝑜 ∅))
19 eqimss 3690 . . . . 5 ((𝐴𝑜 ∅) = (𝐵𝑜 ∅) → (𝐴𝑜 ∅) ⊆ (𝐵𝑜 ∅))
2018, 19syl 17 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 ∅) ⊆ (𝐵𝑜 ∅))
21 simpl 472 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ On)
22 onelss 5804 . . . . . . 7 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
2322imp 444 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴𝐵)
2413, 21, 23jca31 556 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵))
25 oecl 7662 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
26253adant2 1100 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
27 oecl 7662 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 𝑦) ∈ On)
28273adant1 1099 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 𝑦) ∈ On)
29 simp1 1081 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30 omwordri 7697 . . . . . . . . . . . . 13 (((𝐴𝑜 𝑦) ∈ On ∧ (𝐵𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐴)))
3126, 28, 29, 30syl3anc 1366 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐴)))
3231imp 444 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦)) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐴))
3332adantrl 752 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐴))
34 omwordi 7696 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵𝑜 𝑦) ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵)))
3528, 34syld3an3 1411 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵)))
3635imp 444 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝐵) → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵))
3736adantrr 753 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵))
3833, 37sstrd 3646 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵))
39 oesuc 7652 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
40393adant2 1100 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
4140adantr 480 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
42 oesuc 7652 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 suc 𝑦) = ((𝐵𝑜 𝑦) ·𝑜 𝐵))
43423adant1 1099 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 suc 𝑦) = ((𝐵𝑜 𝑦) ·𝑜 𝐵))
4443adantr 480 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → (𝐵𝑜 suc 𝑦) = ((𝐵𝑜 𝑦) ·𝑜 𝐵))
4538, 41, 443sstr4d 3681 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → (𝐴𝑜 suc 𝑦) ⊆ (𝐵𝑜 suc 𝑦))
4645exp520 1310 . . . . . . 7 (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴𝐵 → ((𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 suc 𝑦) ⊆ (𝐵𝑜 suc 𝑦))))))
4746com3r 87 . . . . . 6 (𝑦 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → ((𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 suc 𝑦) ⊆ (𝐵𝑜 suc 𝑦))))))
4847imp4c 616 . . . . 5 (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ((𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 suc 𝑦) ⊆ (𝐵𝑜 suc 𝑦))))
4924, 48syl5 34 . . . 4 (𝑦 ∈ On → ((𝐵 ∈ On ∧ 𝐴𝐵) → ((𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 suc 𝑦) ⊆ (𝐵𝑜 suc 𝑦))))
50 vex 3234 . . . . . . . . . . . 12 𝑥 ∈ V
51 limelon 5826 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5250, 51mpan 706 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
53 0ellim 5825 . . . . . . . . . . 11 (Lim 𝑥 → ∅ ∈ 𝑥)
54 oe0m1 7646 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑𝑜 𝑥) = ∅))
5554biimpa 500 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑𝑜 𝑥) = ∅)
5652, 53, 55syl2anc 694 . . . . . . . . . 10 (Lim 𝑥 → (∅ ↑𝑜 𝑥) = ∅)
57 0ss 4005 . . . . . . . . . 10 ∅ ⊆ (𝐵𝑜 𝑥)
5856, 57syl6eqss 3688 . . . . . . . . 9 (Lim 𝑥 → (∅ ↑𝑜 𝑥) ⊆ (𝐵𝑜 𝑥))
59 oveq1 6697 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴𝑜 𝑥) = (∅ ↑𝑜 𝑥))
6059sseq1d 3665 . . . . . . . . 9 (𝐴 = ∅ → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (∅ ↑𝑜 𝑥) ⊆ (𝐵𝑜 𝑥)))
6158, 60syl5ibr 236 . . . . . . . 8 (𝐴 = ∅ → (Lim 𝑥 → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥)))
6261adantl 481 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥)))
6362a1dd 50 . . . . . 6 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥))))
64 ss2iun 4568 . . . . . . . 8 (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → 𝑦𝑥 (𝐴𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵𝑜 𝑦))
65 oelim 7659 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6650, 65mpanlr1 722 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6766an32s 863 . . . . . . . . . 10 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6867adantllr 755 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6921anim1i 591 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵 ∈ On ∧ Lim 𝑥))
70 ne0i 3954 . . . . . . . . . . . . . . 15 (𝐴𝐵𝐵 ≠ ∅)
71 on0eln0 5818 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
7270, 71syl5ibr 236 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
7372imp 444 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐴𝐵) → ∅ ∈ 𝐵)
7473adantr 480 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75 oelim 7659 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7650, 75mpanlr1 722 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7769, 74, 76syl2anc 694 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7877adantlr 751 . . . . . . . . . 10 ((((𝐵 ∈ On ∧ 𝐴𝐵) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7978adantlll 754 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
8068, 79sseq12d 3667 . . . . . . . 8 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ 𝑦𝑥 (𝐴𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵𝑜 𝑦)))
8164, 80syl5ibr 236 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥)))
8281ex 449 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥))))
8363, 82oe0lem 7638 . . . . 5 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥))))
8413ancri 574 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)))
8583, 84syl11 33 . . . 4 (Lim 𝑥 → ((𝐵 ∈ On ∧ 𝐴𝐵) → (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥))))
863, 6, 9, 12, 20, 49, 85tfinds3 7106 . . 3 (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
8786expd 451 . 2 (𝐶 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶))))
8887impcom 445 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  wss 3607  c0 3948   ciun 4552  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690  1𝑜c1o 7598   ·𝑜 comu 7603  𝑜 coe 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-oexp 7611
This theorem is referenced by:  oeordsuc  7719
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