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Mirrors > Home > MPE Home > Th. List > omwordi | Structured version Visualization version GIF version |
Description: Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.) |
Ref | Expression |
---|---|
omwordi | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omword 7804 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))) | |
2 | 1 | biimpd 219 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))) |
3 | 2 | ex 397 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))) |
4 | eloni 5876 | . . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
5 | ord0eln0 5922 | . . . . . . 7 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅)) | |
6 | 5 | necon2bbid 2986 | . . . . . 6 ⊢ (Ord 𝐶 → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶)) |
7 | 4, 6 | syl 17 | . . . . 5 ⊢ (𝐶 ∈ On → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶)) |
8 | 7 | 3ad2ant3 1129 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶)) |
9 | ssid 3773 | . . . . . . 7 ⊢ ∅ ⊆ ∅ | |
10 | om0r 7773 | . . . . . . . . 9 ⊢ (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅) | |
11 | 10 | adantr 466 | . . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐴) = ∅) |
12 | om0r 7773 | . . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅) | |
13 | 12 | adantl 467 | . . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐵) = ∅) |
14 | 11, 13 | sseq12d 3783 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵) ↔ ∅ ⊆ ∅)) |
15 | 9, 14 | mpbiri 248 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵)) |
16 | oveq1 6800 | . . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) = (∅ ·𝑜 𝐴)) | |
17 | oveq1 6800 | . . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵)) | |
18 | 16, 17 | sseq12d 3783 | . . . . . 6 ⊢ (𝐶 = ∅ → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ (∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵))) |
19 | 15, 18 | syl5ibrcom 237 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))) |
20 | 19 | 3adant3 1126 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))) |
21 | 8, 20 | sylbird 250 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))) |
22 | 21 | a1dd 50 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))) |
23 | 3, 22 | pm2.61d 171 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ∅c0 4063 Ord word 5865 Oncon0 5866 (class class class)co 6793 ·𝑜 comu 7711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-oadd 7717 df-omul 7718 |
This theorem is referenced by: omword1 7807 omass 7814 omeulem1 7816 oewordri 7826 oeoalem 7830 oeeui 7836 oaabs2 7879 omxpenlem 8217 cantnflt 8733 cantnflem1d 8749 |
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