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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 6108 | . . . 4 ⊢ Rel ◡𝐴 | |
2 | ctex 8983 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
3 | cnvexg 7932 | . . . . 5 ⊢ (𝐴 ∈ V → ◡𝐴 ∈ V) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐴 ≼ ω → ◡𝐴 ∈ V) |
5 | cnven 9057 | . . . 4 ⊢ ((Rel ◡𝐴 ∧ ◡𝐴 ∈ V) → ◡𝐴 ≈ ◡◡𝐴) | |
6 | 1, 4, 5 | sylancr 586 | . . 3 ⊢ (𝐴 ≼ ω → ◡𝐴 ≈ ◡◡𝐴) |
7 | cnvcnvss 6198 | . . . 4 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
8 | ssdomg 9020 | . . . 4 ⊢ (𝐴 ∈ V → (◡◡𝐴 ⊆ 𝐴 → ◡◡𝐴 ≼ 𝐴)) | |
9 | 2, 7, 8 | mpisyl 21 | . . 3 ⊢ (𝐴 ≼ ω → ◡◡𝐴 ≼ 𝐴) |
10 | endomtr 9032 | . . 3 ⊢ ((◡𝐴 ≈ ◡◡𝐴 ∧ ◡◡𝐴 ≼ 𝐴) → ◡𝐴 ≼ 𝐴) | |
11 | 6, 9, 10 | syl2anc 583 | . 2 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ 𝐴) |
12 | domtr 9027 | . 2 ⊢ ((◡𝐴 ≼ 𝐴 ∧ 𝐴 ≼ ω) → ◡𝐴 ≼ ω) | |
13 | 11, 12 | mpancom 687 | 1 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3471 ⊆ wss 3947 class class class wbr 5148 ◡ccnv 5677 Rel wrel 5683 ωcom 7870 ≈ cen 8960 ≼ cdom 8961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-1st 7993 df-2nd 7994 df-en 8964 df-dom 8965 |
This theorem is referenced by: rnct 10548 |
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