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Mirrors > Home > MPE Home > Th. List > ctex | Structured version Visualization version GIF version |
Description: A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Ref | Expression |
---|---|
ctex | ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 7852 | . 2 ⊢ (𝐴 ≼ ω → ∃𝑓 𝑓:𝐴–1-1→ω) | |
2 | f1dm 6018 | . . . 4 ⊢ (𝑓:𝐴–1-1→ω → dom 𝑓 = 𝐴) | |
3 | vex 3176 | . . . . 5 ⊢ 𝑓 ∈ V | |
4 | 3 | dmex 6991 | . . . 4 ⊢ dom 𝑓 ∈ V |
5 | 2, 4 | syl6eqelr 2697 | . . 3 ⊢ (𝑓:𝐴–1-1→ω → 𝐴 ∈ V) |
6 | 5 | exlimiv 1845 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→ω → 𝐴 ∈ V) |
7 | 1, 6 | syl 17 | 1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 dom cdm 5038 –1-1→wf1 5801 ωcom 6957 ≼ cdom 7839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-fn 5807 df-f 5808 df-f1 5809 df-dom 7843 |
This theorem is referenced by: ssct 7926 xpct 8722 fimact 9238 cctop 20620 fnct 28876 dmct 28877 cnvct 28878 mptct 28880 mptctf 28883 elsigagen2 29538 measvunilem 29602 measvunilem0 29603 measvuni 29604 sxbrsigalem1 29674 omssubadd 29689 carsggect 29707 pmeasadd 29714 mpct 38388 axccdom 38411 |
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