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Mirrors > Home > MPE Home > Th. List > ssct | Structured version Visualization version GIF version |
Description: Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
Ref | Expression |
---|---|
ssct | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctex 8211 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
2 | ssdomg 8242 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐵 ≼ ω → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) |
4 | 3 | impcom 397 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ 𝐵) |
5 | domtr 8249 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) | |
6 | 4, 5 | sylancom 583 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 Vcvv 3386 ⊆ wss 3770 class class class wbr 4844 ωcom 7300 ≼ cdom 8194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-dom 8198 |
This theorem is referenced by: measvuni 30792 measiuns 30795 sxbrsigalem1 30862 ssnct 40003 fzct 40335 fzoct 40342 salexct 41290 opnvonmbllem2 41588 |
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