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Mirrors > Home > MPE Home > Th. List > oeword | Structured version Visualization version GIF version |
Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
Ref | Expression |
---|---|
oeword | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oeord 7713 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵))) | |
2 | oecan 7714 | . . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ↔ 𝐴 = 𝐵)) | |
3 | 2 | 3coml 1292 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ↔ 𝐴 = 𝐵)) |
4 | 3 | bicomd 213 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 = 𝐵 ↔ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵))) |
5 | 1, 4 | orbi12d 746 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵)))) |
6 | onsseleq 5803 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
7 | 6 | 3adant3 1101 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
8 | eldifi 3765 | . . . 4 ⊢ (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On) | |
9 | id 22 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On)) | |
10 | oecl 7662 | . . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ On) | |
11 | oecl 7662 | . . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑𝑜 𝐵) ∈ On) | |
12 | 10, 11 | anim12dan 900 | . . . . 5 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ↑𝑜 𝐴) ∈ On ∧ (𝐶 ↑𝑜 𝐵) ∈ On)) |
13 | onsseleq 5803 | . . . . 5 ⊢ (((𝐶 ↑𝑜 𝐴) ∈ On ∧ (𝐶 ↑𝑜 𝐵) ∈ On) → ((𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵) ↔ ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵)))) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵) ↔ ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵)))) |
15 | 8, 9, 14 | syl2anr 494 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵) ↔ ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵)))) |
16 | 15 | 3impa 1278 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵) ↔ ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵)))) |
17 | 5, 7, 16 | 3bitr4d 300 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∖ cdif 3604 ⊆ wss 3607 Oncon0 5761 (class class class)co 6690 2𝑜c2o 7599 ↑𝑜 coe 7604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-omul 7610 df-oexp 7611 |
This theorem is referenced by: oewordi 7716 |
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