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Mirrors > Home > MPE Home > Th. List > omword1 | Structured version Visualization version GIF version |
Description: An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.) |
Ref | Expression |
---|---|
omword1 | ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·𝑜 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 5875 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | ordgt0ge1 7735 | . . . . 5 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1𝑜 ⊆ 𝐵)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 1𝑜 ⊆ 𝐵)) |
4 | 3 | adantl 467 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 1𝑜 ⊆ 𝐵)) |
5 | 1on 7724 | . . . . . 6 ⊢ 1𝑜 ∈ On | |
6 | omwordi 7809 | . . . . . 6 ⊢ ((1𝑜 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜 ⊆ 𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵))) | |
7 | 5, 6 | mp3an1 1559 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜 ⊆ 𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵))) |
8 | 7 | ancoms 446 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜 ⊆ 𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵))) |
9 | om1 7780 | . . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 1𝑜) = 𝐴) | |
10 | 9 | adantr 466 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 1𝑜) = 𝐴) |
11 | 10 | sseq1d 3781 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵) ↔ 𝐴 ⊆ (𝐴 ·𝑜 𝐵))) |
12 | 8, 11 | sylibd 229 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜 ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ·𝑜 𝐵))) |
13 | 4, 12 | sylbid 230 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 → 𝐴 ⊆ (𝐴 ·𝑜 𝐵))) |
14 | 13 | imp 393 | 1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·𝑜 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ∅c0 4063 Ord word 5864 Oncon0 5865 (class class class)co 6796 1𝑜c1o 7710 ·𝑜 comu 7715 |
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 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
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 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-omul 7722 |
This theorem is referenced by: om00 7813 cantnflem3 8756 cantnflem4 8757 cnfcomlem 8764 |
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