Theorem ctex 7970

Description: A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.)

Assertion

Ref	Expression
ctex	⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Proof of Theorem ctex

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 7966	. 2 ⊢ (𝐴 ≼ ω → ∃𝑓 𝑓:𝐴–1-1→ω)
2		f1dm 6105	. . . 4 ⊢ (𝑓:𝐴–1-1→ω → dom 𝑓 = 𝐴)
3		vex 3203	. . . . 5 ⊢ 𝑓 ∈ V
4	3	dmex 7099	. . . 4 ⊢ dom 𝑓 ∈ V
5	2, 4	syl6eqelr 2710	. . 3 ⊢ (𝑓:𝐴–1-1→ω → 𝐴 ∈ V)
6	5	exlimiv 1858	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→ω → 𝐴 ∈ V)
7	1, 6	syl 17	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   class class class wbr 4653  dom cdm 5114  –1-1→wf1 5885  ωcom 7065   ≼ cdom 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-fn 5891  df-f 5892  df-f1 5893  df-dom 7957

This theorem is referenced by:  cnvct  8033  ssct  8041  xpct  8839  dmct  9346  fimact  9357  fnct  9359  mptct  9360  cctop  20810  mptctf  29495  elsigagen2  30211  measvunilem  30275  measvunilem0  30276  measvuni  30277  sxbrsigalem1  30347  omssubadd  30362  carsggect  30380  pmeasadd  30387  mpct  39393  axccdom  39416