| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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.) (Proof shortened by Jim Kingdon, 13-Mar-2023.) |
| Ref | Expression |
|---|---|
| ctex | ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reldom 8991 | . 2 ⊢ Rel ≼ | |
| 2 | 1 | brrelex1i 5741 | 1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ωcom 7887 ≼ 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dom 8987 |
| This theorem is referenced by: cnvct 9074 ssctOLD 9092 xpct 10056 iunfictbso 10154 unctb 10244 dmct 10564 fimact 10575 fnct 10577 mptct 10578 iunctb 10614 cctop 23013 1stcrestlem 23460 2ndcdisj2 23465 dis2ndc 23468 uniiccdif 25613 mptctf 32729 elsigagen2 34149 measvunilem 34213 measvunilem0 34214 measvuni 34215 sxbrsigalem1 34287 omssubadd 34302 carsggect 34320 pmeasadd 34327 mpct 45206 axccdom 45227 rn1st 45280 |
| Copyright terms: Public domain | W3C validator |