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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 8227 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) | |
2 | 1 | biimpd 232 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) |
3 | 2 | ex 416 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))) |
4 | eloni 6182 | . . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
5 | ord0eln0 6226 | . . . . . . 7 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅)) | |
6 | 5 | necon2bbid 2977 | . . . . . 6 ⊢ (Ord 𝐶 → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶)) |
7 | 4, 6 | syl 17 | . . . . 5 ⊢ (𝐶 ∈ On → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶)) |
8 | 7 | 3ad2ant3 1136 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶)) |
9 | ssid 3899 | . . . . . . 7 ⊢ ∅ ⊆ ∅ | |
10 | om0r 8195 | . . . . . . . . 9 ⊢ (𝐴 ∈ On → (∅ ·o 𝐴) = ∅) | |
11 | 10 | adantr 484 | . . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐴) = ∅) |
12 | om0r 8195 | . . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ·o 𝐵) = ∅) | |
13 | 12 | adantl 485 | . . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐵) = ∅) |
14 | 11, 13 | sseq12d 3910 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((∅ ·o 𝐴) ⊆ (∅ ·o 𝐵) ↔ ∅ ⊆ ∅)) |
15 | 9, 14 | mpbiri 261 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵)) |
16 | oveq1 7177 | . . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ·o 𝐴) = (∅ ·o 𝐴)) | |
17 | oveq1 7177 | . . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ·o 𝐵) = (∅ ·o 𝐵)) | |
18 | 16, 17 | sseq12d 3910 | . . . . . 6 ⊢ (𝐶 = ∅ → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵))) |
19 | 15, 18 | syl5ibrcom 250 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 = ∅ → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) |
20 | 19 | 3adant3 1133 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) |
21 | 8, 20 | sylbird 263 | . . 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 182 | 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 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ⊆ wss 3843 ∅c0 4211 Ord word 6171 Oncon0 6172 (class class class)co 7170 ·o comu 8129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-oadd 8135 df-omul 8136 |
This theorem is referenced by: omword1 8230 omass 8237 omeulem1 8239 oewordri 8249 oeoalem 8253 oeeui 8259 oaabs2 8303 omxpenlem 8667 cantnflt 9208 cantnflem1d 9224 |
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