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Mirrors > Home > MPE Home > Th. List > cnvct | Structured version Visualization version GIF version |
Description: If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Ref | Expression |
---|---|
cnvct | ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5721 | . . . 4 ⊢ Rel ◡𝐴 | |
2 | ctex 8211 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
3 | cnvexg 7348 | . . . . 5 ⊢ (𝐴 ∈ V → ◡𝐴 ∈ V) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐴 ≼ ω → ◡𝐴 ∈ V) |
5 | cnven 8272 | . . . 4 ⊢ ((Rel ◡𝐴 ∧ ◡𝐴 ∈ V) → ◡𝐴 ≈ ◡◡𝐴) | |
6 | 1, 4, 5 | sylancr 582 | . . 3 ⊢ (𝐴 ≼ ω → ◡𝐴 ≈ ◡◡𝐴) |
7 | cnvcnvss 5806 | . . . 4 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
8 | ssdomg 8242 | . . . 4 ⊢ (𝐴 ∈ V → (◡◡𝐴 ⊆ 𝐴 → ◡◡𝐴 ≼ 𝐴)) | |
9 | 2, 7, 8 | mpisyl 21 | . . 3 ⊢ (𝐴 ≼ ω → ◡◡𝐴 ≼ 𝐴) |
10 | endomtr 8254 | . . 3 ⊢ ((◡𝐴 ≈ ◡◡𝐴 ∧ ◡◡𝐴 ≼ 𝐴) → ◡𝐴 ≼ 𝐴) | |
11 | 6, 9, 10 | syl2anc 580 | . 2 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ 𝐴) |
12 | domtr 8249 | . 2 ⊢ ((◡𝐴 ≼ 𝐴 ∧ 𝐴 ≼ ω) → ◡𝐴 ≼ ω) | |
13 | 11, 12 | mpancom 680 | 1 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 Vcvv 3386 ⊆ wss 3770 class class class wbr 4844 ◡ccnv 5312 Rel wrel 5318 ωcom 7300 ≈ cen 8193 ≼ 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-ne 2973 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 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-mpt 4924 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-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-1st 7402 df-2nd 7403 df-en 8197 df-dom 8198 |
This theorem is referenced by: rnct 9636 |
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