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Theorem oewordri 8539
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 7365 . . . . 5 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
2 oveq2 7365 . . . . 5 (𝑥 = ∅ → (𝐵o 𝑥) = (𝐵o ∅))
31, 2sseq12d 3977 . . . 4 (𝑥 = ∅ → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o ∅) ⊆ (𝐵o ∅)))
4 oveq2 7365 . . . . 5 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
5 oveq2 7365 . . . . 5 (𝑥 = 𝑦 → (𝐵o 𝑥) = (𝐵o 𝑦))
64, 5sseq12d 3977 . . . 4 (𝑥 = 𝑦 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o 𝑦) ⊆ (𝐵o 𝑦)))
7 oveq2 7365 . . . . 5 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
8 oveq2 7365 . . . . 5 (𝑥 = suc 𝑦 → (𝐵o 𝑥) = (𝐵o suc 𝑦))
97, 8sseq12d 3977 . . . 4 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦)))
10 oveq2 7365 . . . . 5 (𝑥 = 𝐶 → (𝐴o 𝑥) = (𝐴o 𝐶))
11 oveq2 7365 . . . . 5 (𝑥 = 𝐶 → (𝐵o 𝑥) = (𝐵o 𝐶))
1210, 11sseq12d 3977 . . . 4 (𝑥 = 𝐶 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
13 onelon 6342 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
14 oe0 8468 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) = 1o)
1513, 14syl 17 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) = 1o)
16 oe0 8468 . . . . . . 7 (𝐵 ∈ On → (𝐵o ∅) = 1o)
1716adantr 481 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐵o ∅) = 1o)
1815, 17eqtr4d 2779 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) = (𝐵o ∅))
19 eqimss 4000 . . . . 5 ((𝐴o ∅) = (𝐵o ∅) → (𝐴o ∅) ⊆ (𝐵o ∅))
2018, 19syl 17 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) ⊆ (𝐵o ∅))
21 simpl 483 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ On)
22 onelss 6359 . . . . . . 7 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
2322imp 407 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴𝐵)
2413, 21, 23jca31 515 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵))
25 oecl 8483 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
26253adant2 1131 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
27 oecl 8483 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o 𝑦) ∈ On)
28273adant1 1130 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o 𝑦) ∈ On)
29 simp1 1136 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30 omwordri 8519 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝑦) ⊆ (𝐵o 𝑦) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴)))
3231imp 407 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦)) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴))
3332adantrl 714 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴))
34 omwordi 8518 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵o 𝑦) ∈ On) → (𝐴𝐵 → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵)))
3528, 34syld3an3 1409 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝐵 → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵)))
3635imp 407 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝐵) → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
3736adantrr 715 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
3833, 37sstrd 3954 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
39 oesuc 8473 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
40393adant2 1131 . . . . . . . . . 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 8473 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o suc 𝑦) = ((𝐵o 𝑦) ·o 𝐵))
43423adant1 1130 . . . . . . . . . 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 3991 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦))
4645exp520 1357 . . . . . . 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 424 . . . . 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 3449 . . . . . . . . . . . 12 𝑥 ∈ V
51 limelon 6381 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5250, 51mpan 688 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
53 0ellim 6380 . . . . . . . . . . 11 (Lim 𝑥 → ∅ ∈ 𝑥)
54 oe0m1 8467 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑o 𝑥) = ∅))
5554biimpa 477 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑o 𝑥) = ∅)
5652, 53, 55syl2anc 584 . . . . . . . . . 10 (Lim 𝑥 → (∅ ↑o 𝑥) = ∅)
57 0ss 4356 . . . . . . . . . 10 ∅ ⊆ (𝐵o 𝑥)
5856, 57eqsstrdi 3998 . . . . . . . . 9 (Lim 𝑥 → (∅ ↑o 𝑥) ⊆ (𝐵o 𝑥))
59 oveq1 7364 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴o 𝑥) = (∅ ↑o 𝑥))
6059sseq1d 3975 . . . . . . . . 9 (𝐴 = ∅ → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (∅ ↑o 𝑥) ⊆ (𝐵o 𝑥)))
6158, 60syl5ibr 245 . . . . . . . 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 4972 . . . . . . . 8 (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → 𝑦𝑥 (𝐴o 𝑦) ⊆ 𝑦𝑥 (𝐵o 𝑦))
65 oelim 8480 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6650, 65mpanlr1 704 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6766an32s 650 . . . . . . . . . 10 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6867adantllr 717 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6921anim1i 615 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵 ∈ On ∧ Lim 𝑥))
70 ne0i 4294 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
71 on0eln0 6373 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
7270, 71syl5ibr 245 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
7372imp 407 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐴𝐵) → ∅ ∈ 𝐵)
7473adantr 481 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75 oelim 8480 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7650, 75mpanlr1 704 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7769, 74, 76syl2anc 584 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7877ad4ant24 752 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7968, 78sseq12d 3977 . . . . . . . 8 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ 𝑦𝑥 (𝐴o 𝑦) ⊆ 𝑦𝑥 (𝐵o 𝑦)))
8064, 79syl5ibr 245 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥)))
8180ex 413 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
8263, 81oe0lem 8459 . . . . 5 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
8313ancri 550 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)))
8482, 83syl11 33 . . . 4 (Lim 𝑥 → ((𝐵 ∈ On ∧ 𝐴𝐵) → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
853, 6, 9, 12, 20, 49, 84tfinds3 7801 . . 3 (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
8685expd 416 . 2 (𝐶 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶))))
8786impcom 408 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  wss 3910  c0 4282   ciun 4954  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7357  1oc1o 8405   ·o comu 8410  o coe 8411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-oexp 8418
This theorem is referenced by:  oeordsuc  8541  oege2  41627
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