Theorem el0c 4422

Description: Membership in cardinal zero. (Contributed by SF, 22-Jan-2015.)

Assertion

Ref	Expression
el0c	⊢ (A ∈ 0c ↔ A = ∅)

Proof of Theorem el0c

Step	Hyp	Ref	Expression
1		df-0c 4378	. . 3 ⊢ 0c = {∅}
2	1	eleq2i 2417	. 2 ⊢ (A ∈ 0c ↔ A ∈ {∅})
3		0ex 4111	. . 3 ⊢ ∅ ∈ V
4	3	elsnc2 3763	. 2 ⊢ (A ∈ {∅} ↔ A = ∅)
5	2, 4	bitri 240	1 ⊢ (A ∈ 0c ↔ A = ∅)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∅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

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-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:  nulel0c  4423  0fin  4424  ncfinraise  4482  ncfinlower  4484  nnadjoin  4521  nnpweq  4524  sfin01  4529  tfinnn  4535  sfinltfin  4536  vfin1cltv  4548  df0c2  6138  ncvsq  6257  0lt1c  6259  nmembers1lem2  6270