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 8503 | . 2 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 5601 | 1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 class class class wbr 5057 ωcom 7569 ≼ cdom 8495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-dom 8499 |
This theorem is referenced by: cnvct 8574 ssct 8586 xpct 9430 iunfictbso 9528 unctb 9615 dmct 9934 fimact 9945 fnct 9947 mptct 9948 iunctb 9984 cctop 21542 1stcrestlem 21988 2ndcdisj2 21993 dis2ndc 21996 uniiccdif 24106 mptctf 30379 elsigagen2 31306 measvunilem 31370 measvunilem0 31371 measvuni 31372 sxbrsigalem1 31442 omssubadd 31457 carsggect 31475 pmeasadd 31482 mpct 41340 axccdom 41363 |
Copyright terms: Public domain | W3C validator |