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Mirrors > Home > MPE Home > Th. List > omword | Structured version Visualization version GIF version |
Description: Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.) |
Ref | Expression |
---|---|
omword | ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omord2 8588 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))) | |
2 | 3anrot 1097 | . . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ↔ (𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On)) | |
3 | omcan 8590 | . . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ↔ 𝐴 = 𝐵)) | |
4 | 2, 3 | sylanbr 580 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ↔ 𝐴 = 𝐵)) |
5 | 4 | bicomd 222 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 ↔ (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵))) |
6 | 1, 5 | orbi12d 916 | . 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵)))) |
7 | onsseleq 6412 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
8 | 7 | 3adant3 1129 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
9 | 8 | adantr 479 | . 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
10 | omcl 8557 | . . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o 𝐴) ∈ On) | |
11 | omcl 8557 | . . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ·o 𝐵) ∈ On) | |
12 | 10, 11 | anim12dan 617 | . . . . . 6 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On)) |
13 | 12 | ancoms 457 | . . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On)) |
14 | 13 | 3impa 1107 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On)) |
15 | onsseleq 6412 | . . . 4 ⊢ (((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)))) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵)))) |
17 | 16 | adantr 479 | . 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵)))) |
18 | 6, 9, 17 | 3bitr4d 310 | 1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ∅c0 4322 Oncon0 6371 (class class class)co 7419 ·o comu 8485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-oadd 8491 df-omul 8492 |
This theorem is referenced by: omwordi 8592 omeulem2 8604 oeeui 8623 |
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