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Theorem oewordri 8628
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 7438 . . . . 5 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
2 oveq2 7438 . . . . 5 (𝑥 = ∅ → (𝐵o 𝑥) = (𝐵o ∅))
31, 2sseq12d 4028 . . . 4 (𝑥 = ∅ → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o ∅) ⊆ (𝐵o ∅)))
4 oveq2 7438 . . . . 5 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
5 oveq2 7438 . . . . 5 (𝑥 = 𝑦 → (𝐵o 𝑥) = (𝐵o 𝑦))
64, 5sseq12d 4028 . . . 4 (𝑥 = 𝑦 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o 𝑦) ⊆ (𝐵o 𝑦)))
7 oveq2 7438 . . . . 5 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
8 oveq2 7438 . . . . 5 (𝑥 = suc 𝑦 → (𝐵o 𝑥) = (𝐵o suc 𝑦))
97, 8sseq12d 4028 . . . 4 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦)))
10 oveq2 7438 . . . . 5 (𝑥 = 𝐶 → (𝐴o 𝑥) = (𝐴o 𝐶))
11 oveq2 7438 . . . . 5 (𝑥 = 𝐶 → (𝐵o 𝑥) = (𝐵o 𝐶))
1210, 11sseq12d 4028 . . . 4 (𝑥 = 𝐶 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
13 onelon 6410 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
14 oe0 8558 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) = 1o)
1513, 14syl 17 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) = 1o)
16 oe0 8558 . . . . . . 7 (𝐵 ∈ On → (𝐵o ∅) = 1o)
1716adantr 480 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐵o ∅) = 1o)
1815, 17eqtr4d 2777 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) = (𝐵o ∅))
19 eqimss 4053 . . . . 5 ((𝐴o ∅) = (𝐵o ∅) → (𝐴o ∅) ⊆ (𝐵o ∅))
2018, 19syl 17 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) ⊆ (𝐵o ∅))
21 simpl 482 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ On)
22 onelss 6427 . . . . . . 7 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
2322imp 406 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴𝐵)
2413, 21, 23jca31 514 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵))
25 oecl 8573 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
26253adant2 1130 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
27 oecl 8573 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o 𝑦) ∈ On)
28273adant1 1129 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o 𝑦) ∈ On)
29 simp1 1135 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30 omwordri 8608 . . . . . . . . . . . . 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 1370 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝑦) ⊆ (𝐵o 𝑦) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴)))
3231imp 406 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦)) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴))
3332adantrl 716 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴))
34 omwordi 8607 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵o 𝑦) ∈ On) → (𝐴𝐵 → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵)))
3528, 34syld3an3 1408 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝐵 → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵)))
3635imp 406 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝐵) → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
3736adantrr 717 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
3833, 37sstrd 4005 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
39 oesuc 8563 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
40393adant2 1130 . . . . . . . . . 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 8563 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o suc 𝑦) = ((𝐵o 𝑦) ·o 𝐵))
43423adant1 1129 . . . . . . . . . 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 4042 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦))
4645exp520 1356 . . . . . . 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 423 . . . . 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 3481 . . . . . . . . . . . 12 𝑥 ∈ V
51 limelon 6449 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5250, 51mpan 690 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
53 0ellim 6448 . . . . . . . . . . 11 (Lim 𝑥 → ∅ ∈ 𝑥)
54 oe0m1 8557 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑o 𝑥) = ∅))
5554biimpa 476 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑o 𝑥) = ∅)
5652, 53, 55syl2anc 584 . . . . . . . . . 10 (Lim 𝑥 → (∅ ↑o 𝑥) = ∅)
57 0ss 4405 . . . . . . . . . 10 ∅ ⊆ (𝐵o 𝑥)
5856, 57eqsstrdi 4049 . . . . . . . . 9 (Lim 𝑥 → (∅ ↑o 𝑥) ⊆ (𝐵o 𝑥))
59 oveq1 7437 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴o 𝑥) = (∅ ↑o 𝑥))
6059sseq1d 4026 . . . . . . . . 9 (𝐴 = ∅ → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (∅ ↑o 𝑥) ⊆ (𝐵o 𝑥)))
6158, 60imbitrrid 246 . . . . . . . 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 5014 . . . . . . . 8 (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → 𝑦𝑥 (𝐴o 𝑦) ⊆ 𝑦𝑥 (𝐵o 𝑦))
65 oelim 8570 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6650, 65mpanlr1 706 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6766an32s 652 . . . . . . . . . 10 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6867adantllr 719 . . . . . . . . 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 4346 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
71 on0eln0 6441 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
7270, 71imbitrrid 246 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
7372imp 406 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐴𝐵) → ∅ ∈ 𝐵)
7473adantr 480 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75 oelim 8570 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7650, 75mpanlr1 706 . . . . . . . . . . 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 754 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7968, 78sseq12d 4028 . . . . . . . 8 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ 𝑦𝑥 (𝐴o 𝑦) ⊆ 𝑦𝑥 (𝐵o 𝑦)))
8064, 79imbitrrid 246 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥)))
8180ex 412 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
8263, 81oe0lem 8549 . . . . 5 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
8313ancri 549 . . . . 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 7885 . . 3 (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
8685expd 415 . 2 (𝐶 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶))))
8786impcom 407 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  wss 3962  c0 4338   ciun 4995  Oncon0 6385  Lim wlim 6386  suc csuc 6387  (class class class)co 7430  1oc1o 8497   ·o comu 8502  o coe 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-omul 8509  df-oexp 8510
This theorem is referenced by:  oeordsuc  8630  oege2  43296
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