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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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