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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.) Avoid ax-pow 5368, ax-un 7745. (Revised by BTernaryTau, 7-Dec-2024.) |
Ref | Expression |
---|---|
ssct | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domssl 9028 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ⊆ wss 3946 class class class wbr 5152 ωcom 7875 ≼ cdom 8971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-dom 8975 |
This theorem is referenced by: measvuni 34003 measiuns 34006 sxbrsigalem1 34075 ssnct 44615 fzct 44931 fzoct 44936 salexct 45892 opnvonmbllem2 46191 |
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