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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 6122 | . . . 4 ⊢ Rel ◡𝐴 | |
| 2 | ctex 9004 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
| 3 | cnvexg 7946 | . . . . 5 ⊢ (𝐴 ∈ V → ◡𝐴 ∈ V) | |
| 4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐴 ≼ ω → ◡𝐴 ∈ V) |
| 5 | cnven 9073 | . . . 4 ⊢ ((Rel ◡𝐴 ∧ ◡𝐴 ∈ V) → ◡𝐴 ≈ ◡◡𝐴) | |
| 6 | 1, 4, 5 | sylancr 587 | . . 3 ⊢ (𝐴 ≼ ω → ◡𝐴 ≈ ◡◡𝐴) |
| 7 | cnvcnvss 6214 | . . . 4 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
| 8 | ssdomg 9040 | . . . 4 ⊢ (𝐴 ∈ V → (◡◡𝐴 ⊆ 𝐴 → ◡◡𝐴 ≼ 𝐴)) | |
| 9 | 2, 7, 8 | mpisyl 21 | . . 3 ⊢ (𝐴 ≼ ω → ◡◡𝐴 ≼ 𝐴) |
| 10 | endomtr 9052 | . . 3 ⊢ ((◡𝐴 ≈ ◡◡𝐴 ∧ ◡◡𝐴 ≼ 𝐴) → ◡𝐴 ≼ 𝐴) | |
| 11 | 6, 9, 10 | syl2anc 584 | . 2 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ 𝐴) |
| 12 | domtr 9047 | . 2 ⊢ ((◡𝐴 ≼ 𝐴 ∧ 𝐴 ≼ ω) → ◡𝐴 ≼ ω) | |
| 13 | 11, 12 | mpancom 688 | 1 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ◡ccnv 5684 Rel wrel 5690 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1st 8014 df-2nd 8015 df-en 8986 df-dom 8987 |
| This theorem is referenced by: rnct 10565 |
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