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Mirrors > Home > ILE Home > Th. List > nnmword | GIF version |
Description: Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
nnmword | ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iba 300 | . . . 4 ⊢ (∅ ∈ 𝐶 → (𝐵 ∈ 𝐴 ↔ (𝐵 ∈ 𝐴 ∧ ∅ ∈ 𝐶))) | |
2 | nnmord 6531 | . . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵 ∈ 𝐴 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))) | |
3 | 2 | 3com12 1208 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵 ∈ 𝐴 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))) |
4 | 1, 3 | sylan9bbr 463 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 ↔ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))) |
5 | 4 | notbid 668 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (¬ 𝐵 ∈ 𝐴 ↔ ¬ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))) |
6 | simpl1 1001 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐴 ∈ ω) | |
7 | simpl2 1002 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐵 ∈ ω) | |
8 | nntri1 6510 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | |
9 | 6, 7, 8 | syl2anc 411 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) |
10 | simpl3 1003 | . . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ ω) | |
11 | nnmcl 6495 | . . . 4 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·o 𝐴) ∈ ω) | |
12 | 10, 6, 11 | syl2anc 411 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ ω) |
13 | nnmcl 6495 | . . . 4 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·o 𝐵) ∈ ω) | |
14 | 10, 7, 13 | syl2anc 411 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐵) ∈ ω) |
15 | nntri1 6510 | . . 3 ⊢ (((𝐶 ·o 𝐴) ∈ ω ∧ (𝐶 ·o 𝐵) ∈ ω) → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ ¬ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))) | |
16 | 12, 14, 15 | syl2anc 411 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ ¬ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))) |
17 | 5, 9, 16 | 3bitr4d 220 | 1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 979 ∈ wcel 2158 ⊆ wss 3141 ∅c0 3434 ωcom 4601 (class class class)co 5888 ·o comu 6428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-oadd 6434 df-omul 6435 |
This theorem is referenced by: nnmcan 6533 archnqq 7429 |
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