MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oewordri Structured version   Visualization version   GIF version

Theorem oewordri 7826
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 6801 . . . . 5 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
2 oveq2 6801 . . . . 5 (𝑥 = ∅ → (𝐵𝑜 𝑥) = (𝐵𝑜 ∅))
31, 2sseq12d 3783 . . . 4 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 ∅) ⊆ (𝐵𝑜 ∅)))
4 oveq2 6801 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
5 oveq2 6801 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑜 𝑥) = (𝐵𝑜 𝑦))
64, 5sseq12d 3783 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦)))
7 oveq2 6801 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
8 oveq2 6801 . . . . 5 (𝑥 = suc 𝑦 → (𝐵𝑜 𝑥) = (𝐵𝑜 suc 𝑦))
97, 8sseq12d 3783 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 suc 𝑦) ⊆ (𝐵𝑜 suc 𝑦)))
10 oveq2 6801 . . . . 5 (𝑥 = 𝐶 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐶))
11 oveq2 6801 . . . . 5 (𝑥 = 𝐶 → (𝐵𝑜 𝑥) = (𝐵𝑜 𝐶))
1210, 11sseq12d 3783 . . . 4 (𝑥 = 𝐶 → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
13 onelon 5891 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
14 oe0 7756 . . . . . . 7 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
1513, 14syl 17 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 ∅) = 1𝑜)
16 oe0 7756 . . . . . . 7 (𝐵 ∈ On → (𝐵𝑜 ∅) = 1𝑜)
1716adantr 466 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐵𝑜 ∅) = 1𝑜)
1815, 17eqtr4d 2808 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 ∅) = (𝐵𝑜 ∅))
19 eqimss 3806 . . . . 5 ((𝐴𝑜 ∅) = (𝐵𝑜 ∅) → (𝐴𝑜 ∅) ⊆ (𝐵𝑜 ∅))
2018, 19syl 17 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 ∅) ⊆ (𝐵𝑜 ∅))
21 simpl 468 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ On)
22 onelss 5909 . . . . . . 7 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
2322imp 393 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴𝐵)
2413, 21, 23jca31 504 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵))
25 oecl 7771 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
26253adant2 1125 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
27 oecl 7771 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 𝑦) ∈ On)
28273adant1 1124 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 𝑦) ∈ On)
29 simp1 1130 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30 omwordri 7806 . . . . . . . . . . . . 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 1476 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐴)))
3231imp 393 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦)) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐴))
3332adantrl 695 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐴))
34 omwordi 7805 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵𝑜 𝑦) ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵)))
3528, 34syld3an3 1515 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵)))
3635imp 393 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝐵) → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵))
3736adantrr 696 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → ((𝐵𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵))
3833, 37sstrd 3762 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝑦) ·𝑜 𝐵))
39 oesuc 7761 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
40393adant2 1125 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
4140adantr 466 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
42 oesuc 7761 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 suc 𝑦) = ((𝐵𝑜 𝑦) ·𝑜 𝐵))
43423adant1 1124 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑜 suc 𝑦) = ((𝐵𝑜 𝑦) ·𝑜 𝐵))
4443adantr 466 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → (𝐵𝑜 suc 𝑦) = ((𝐵𝑜 𝑦) ·𝑜 𝐵))
4538, 41, 443sstr4d 3797 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦))) → (𝐴𝑜 suc 𝑦) ⊆ (𝐵𝑜 suc 𝑦))
4645exp520 1450 . . . . . . 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 410 . . . . 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 3354 . . . . . . . . . . . 12 𝑥 ∈ V
51 limelon 5931 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5250, 51mpan 670 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
53 0ellim 5930 . . . . . . . . . . 11 (Lim 𝑥 → ∅ ∈ 𝑥)
54 oe0m1 7755 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑𝑜 𝑥) = ∅))
5554biimpa 462 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑𝑜 𝑥) = ∅)
5652, 53, 55syl2anc 573 . . . . . . . . . 10 (Lim 𝑥 → (∅ ↑𝑜 𝑥) = ∅)
57 0ss 4116 . . . . . . . . . 10 ∅ ⊆ (𝐵𝑜 𝑥)
5856, 57syl6eqss 3804 . . . . . . . . 9 (Lim 𝑥 → (∅ ↑𝑜 𝑥) ⊆ (𝐵𝑜 𝑥))
59 oveq1 6800 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴𝑜 𝑥) = (∅ ↑𝑜 𝑥))
6059sseq1d 3781 . . . . . . . . 9 (𝐴 = ∅ → ((𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥) ↔ (∅ ↑𝑜 𝑥) ⊆ (𝐵𝑜 𝑥)))
6158, 60syl5ibr 236 . . . . . . . 8 (𝐴 = ∅ → (Lim 𝑥 → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥)))
6261adantl 467 . . . . . . 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 4670 . . . . . . . 8 (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → 𝑦𝑥 (𝐴𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵𝑜 𝑦))
65 oelim 7768 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6650, 65mpanlr1 686 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6766an32s 631 . . . . . . . . . 10 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6867adantllr 698 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
6921anim1i 602 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵 ∈ On ∧ Lim 𝑥))
70 ne0i 4069 . . . . . . . . . . . . . . 15 (𝐴𝐵𝐵 ≠ ∅)
71 on0eln0 5923 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
7270, 71syl5ibr 236 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
7372imp 393 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐴𝐵) → ∅ ∈ 𝐵)
7473adantr 466 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75 oelim 7768 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7650, 75mpanlr1 686 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7769, 74, 76syl2anc 573 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7877adantlr 694 . . . . . . . . . 10 ((((𝐵 ∈ On ∧ 𝐴𝐵) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
7978adantlll 697 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵𝑜 𝑥) = 𝑦𝑥 (𝐵𝑜 𝑦))
8068, 79sseq12d 3783 . . . . . . . 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 397 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥))))
8363, 82oe0lem 7747 . . . . 5 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴𝑜 𝑦) ⊆ (𝐵𝑜 𝑦) → (𝐴𝑜 𝑥) ⊆ (𝐵𝑜 𝑥))))
8413ancri 539 . . . . 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 7211 . . 3 (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
8786expd 400 . 2 (𝐶 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶))))
8887impcom 394 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  Vcvv 3351  wss 3723  c0 4063   ciun 4654  Oncon0 5866  Lim wlim 5867  suc csuc 5868  (class class class)co 6793  1𝑜c1o 7706   ·𝑜 comu 7711  𝑜 coe 7712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-oexp 7719
This theorem is referenced by:  oeordsuc  7828
  Copyright terms: Public domain W3C validator