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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 6041 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | ordgt0ge1 7926 | . . . . 5 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵)) |
4 | 3 | adantl 474 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵)) |
5 | 1on 7914 | . . . . . 6 ⊢ 1o ∈ On | |
6 | omwordi 8000 | . . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵))) | |
7 | 5, 6 | mp3an1 1427 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵))) |
8 | 7 | ancoms 451 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵))) |
9 | om1 7971 | . . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴) | |
10 | 9 | adantr 473 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 1o) = 𝐴) |
11 | 10 | sseq1d 3890 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵) ↔ 𝐴 ⊆ (𝐴 ·o 𝐵))) |
12 | 8, 11 | sylibd 231 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ·o 𝐵))) |
13 | 4, 12 | sylbid 232 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 → 𝐴 ⊆ (𝐴 ·o 𝐵))) |
14 | 13 | imp 398 | 1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3831 ∅c0 4180 Ord word 6030 Oncon0 6031 (class class class)co 6978 1oc1o 7900 ·o comu 7905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-omul 7912 |
This theorem is referenced by: om00 8004 cantnflem3 8950 cantnflem4 8951 cnfcomlem 8958 |
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