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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 5759 | . . . 4 ⊢ Rel ◡𝐴 | |
2 | ctex 8258 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
3 | cnvexg 7393 | . . . . 5 ⊢ (𝐴 ∈ V → ◡𝐴 ∈ V) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐴 ≼ ω → ◡𝐴 ∈ V) |
5 | cnven 8319 | . . . 4 ⊢ ((Rel ◡𝐴 ∧ ◡𝐴 ∈ V) → ◡𝐴 ≈ ◡◡𝐴) | |
6 | 1, 4, 5 | sylancr 581 | . . 3 ⊢ (𝐴 ≼ ω → ◡𝐴 ≈ ◡◡𝐴) |
7 | cnvcnvss 5844 | . . . 4 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
8 | ssdomg 8289 | . . . 4 ⊢ (𝐴 ∈ V → (◡◡𝐴 ⊆ 𝐴 → ◡◡𝐴 ≼ 𝐴)) | |
9 | 2, 7, 8 | mpisyl 21 | . . 3 ⊢ (𝐴 ≼ ω → ◡◡𝐴 ≼ 𝐴) |
10 | endomtr 8301 | . . 3 ⊢ ((◡𝐴 ≈ ◡◡𝐴 ∧ ◡◡𝐴 ≼ 𝐴) → ◡𝐴 ≼ 𝐴) | |
11 | 6, 9, 10 | syl2anc 579 | . 2 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ 𝐴) |
12 | domtr 8296 | . 2 ⊢ ((◡𝐴 ≼ 𝐴 ∧ 𝐴 ≼ ω) → ◡𝐴 ≼ ω) | |
13 | 11, 12 | mpancom 678 | 1 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 class class class wbr 4888 ◡ccnv 5356 Rel wrel 5362 ωcom 7345 ≈ cen 8240 ≼ cdom 8241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-1st 7447 df-2nd 7448 df-en 8244 df-dom 8245 |
This theorem is referenced by: rnct 9684 |
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