Theorem 0cex 4393

Description: Cardinal zero is a set. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
0cex	⊢ 0c ∈ V

Proof of Theorem 0cex

Step	Hyp	Ref	Expression
1		df-0c 4378	. 2 ⊢ 0c = {∅}
2		snex 4112	. 2 ⊢ {∅} ∈ V
3	1, 2	eqeltri 2423	1 ⊢ 0c ∈ V

Syntax hints:   ∈ wcel 1710  Vcvv 2860  ∅c0 3551  {csn 3738  0_cc0c 4375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-0c 4378

This theorem is referenced by:  peano1  4403  findsd  4411  ltfintri  4467  0ceven  4506  0cnelphi  4598  proj1op  4601  proj2op  4602  dfnnc3  5886  ce0nn  6181  lec0cg  6199  frecxp  6315  dmfrec  6317  fnfreclem2  6319  fnfreclem3  6320