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Theorem oewordri 7624
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 6618 . . . . 5 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
2 oveq2 6618 . . . . 5 (𝑥 = ∅ → (𝐵𝑜 𝑥) = (𝐵𝑜 ∅))
31, 2sseq12d 3618 . . . 4 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 ∅) ⊆ (𝐵𝑜 ∅)))
4 oveq2 6618 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
5 oveq2 6618 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑜 𝑥) = (𝐵𝑜 𝑦))
64, 5sseq12d 3618 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦)))
7 oveq2 6618 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
8 oveq2 6618 . . . . 5 (𝑥 = suc 𝑦 → (𝐵𝑜 𝑥) = (𝐵𝑜 suc 𝑦))
97, 8sseq12d 3618 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 suc 𝑦) ⊆ (𝐵𝑜 suc 𝑦)))
10 oveq2 6618 . . . . 5 (𝑥 = 𝐶 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐶))
11 oveq2 6618 . . . . 5 (𝑥 = 𝐶 → (𝐵𝑜 𝑥) = (𝐵𝑜 𝐶))
1210, 11sseq12d 3618 . . . 4 (𝑥 = 𝐶 → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
13 onelon 5712 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
14 oe0 7554 . . . . . . 7 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
1513, 14syl 17 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 ∅) = 1𝑜)
16 oe0 7554 . . . . . . 7 (𝐵 ∈ On → (𝐵𝑜 ∅) = 1𝑜)
1716adantr 481 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐵𝑜 ∅) = 1𝑜)
1815, 17eqtr4d 2658 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 ∅) = (𝐵𝑜 ∅))
19 eqimss 3641 . . . . 5 ((𝐴𝑜 ∅) = (𝐵𝑜 ∅) → (𝐴𝑜 ∅) ⊆ (𝐵𝑜 ∅))
2018, 19syl 17 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 ∅) ⊆ (𝐵𝑜 ∅))
21 simpl 473 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ On)
22 onelss 5730 . . . . . . 7 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
2322imp 445 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴𝐵)
2413, 21, 23jca31 556 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵))
25 oecl 7569 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
26253adant2 1078 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
27 oecl 7569 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 𝑦) ∈ On)
28273adant1 1077 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 𝑦) ∈ On)
29 simp1 1059 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30 omwordri 7604 . . . . . . . . . . . . 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 1323 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐴)))
3231imp 445 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦)) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐴))
3332adantrl 751 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐴))
34 omwordi 7603 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵𝑜 𝑦) ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵)))
3528, 34syld3an3 1368 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵)))
3635imp 445 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝐵) → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵))
3736adantrr 752 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵))
3833, 37sstrd 3597 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵))
39 oesuc 7559 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
40393adant2 1078 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
4140adantr 481 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
42 oesuc 7559 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 suc 𝑦) = ((𝐵𝑜 𝑦) ·𝑜 𝐵))
43423adant1 1077 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 suc 𝑦) = ((𝐵𝑜 𝑦) ·𝑜 𝐵))
4443adantr 481 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → (𝐵𝑜 suc 𝑦) = ((𝐵𝑜 𝑦) ·𝑜 𝐵))
4538, 41, 443sstr4d 3632 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → (𝐴𝑜 suc 𝑦) ⊆ (𝐵𝑜 suc 𝑦))
4645exp520 1285 . . . . . . 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 3192 . . . . . . . . . . . 12 𝑥 ∈ V
51 limelon 5752 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5250, 51mpan 705 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
53 0ellim 5751 . . . . . . . . . . 11 (Lim 𝑥 → ∅ ∈ 𝑥)
54 oe0m1 7553 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑𝑜 𝑥) = ∅))
5554biimpa 501 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑𝑜 𝑥) = ∅)
5652, 53, 55syl2anc 692 . . . . . . . . . 10 (Lim 𝑥 → (∅ ↑𝑜 𝑥) = ∅)
57 0ss 3949 . . . . . . . . . 10 ∅ ⊆ (𝐵𝑜 𝑥)
5856, 57syl6eqss 3639 . . . . . . . . 9 (Lim 𝑥 → (∅ ↑𝑜 𝑥) ⊆ (𝐵𝑜 𝑥))
59 oveq1 6617 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴𝑜 𝑥) = (∅ ↑𝑜 𝑥))
6059sseq1d 3616 . . . . . . . . 9 (𝐴 = ∅ → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (∅ ↑𝑜 𝑥) ⊆ (𝐵𝑜 𝑥)))
6158, 60syl5ibr 236 . . . . . . . 8 (𝐴 = ∅ → (Lim 𝑥 → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥)))
6261adantl 482 . . . . . . 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 4507 . . . . . . . 8 (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → 𝑦𝑥 (𝐴𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵𝑜 𝑦))
65 oelim 7566 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6650, 65mpanlr1 721 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6766an32s 845 . . . . . . . . . 10 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6867adantllr 754 . . . . . . . . 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 3902 . . . . . . . . . . . . . . 15 (𝐴𝐵𝐵 ≠ ∅)
71 on0eln0 5744 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
7270, 71syl5ibr 236 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
7372imp 445 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐴𝐵) → ∅ ∈ 𝐵)
7473adantr 481 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75 oelim 7566 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7650, 75mpanlr1 721 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7769, 74, 76syl2anc 692 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7877adantlr 750 . . . . . . . . . 10 ((((𝐵 ∈ On ∧ 𝐴𝐵) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7978adantlll 753 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
8068, 79sseq12d 3618 . . . . . . . 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 450 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥))))
8363, 82oe0lem 7545 . . . . 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 7018 . . 3 (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
8786expd 452 . 2 (𝐶 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶))))
8887impcom 446 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3189  wss 3559  c0 3896   ciun 4490  Oncon0 5687  Lim wlim 5688  suc csuc 5689  (class class class)co 6610  1𝑜c1o 7505   ·𝑜 comu 7510  𝑜 coe 7511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-omul 7517  df-oexp 7518
This theorem is referenced by:  oeordsuc  7626
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