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Theorem el0c 4421

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

Assertion

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

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Proof of Theorem el0c

Step	Hyp	Ref	Expression
1		df-0c 4377	. . 3 ⊢ 0c = {∅}
2	1	eleq2i 2417	. 2 ⊢ (A ∈ 0c ↔ A ∈ {∅})
3		0ex 4110	. . 3 ⊢ ∅ ∈ V
4	3	elsnc2 3762	. 2 ⊢ (A ∈ {∅} ↔ A = ∅)
5	2, 4	bitri 240	1 ⊢ (A ∈ 0c ↔ A = ∅)

Colors of variables: wff setvar class

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∅c0 3550  {csn 3737  0cc0c 4374

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-0c 4377

This theorem is referenced by:  nulel0c  4422  0fin  4423  ncfinraise  4481  ncfinlower  4483  nnadjoin  4520  nnpweq  4523  sfin01  4528  tfinnn  4534  sfinltfin  4535  vfin1cltv  4547  df0c2  6137  ncvsq  6256  0lt1c  6258  nmembers1lem2  6269

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