Theorem nulel0c 4423

Description: The empty set is a member of cardinal zero. (Contributed by SF, 13-Feb-2015.)

Assertion

Ref	Expression
nulel0c	⊢ ∅ ∈ 0c

Proof of Theorem nulel0c

Step	Hyp	Ref	Expression
1		eqid 2353	. 2 ⊢ ∅ = ∅
2		el0c 4422	. 2 ⊢ (∅ ∈ 0c ↔ ∅ = ∅)
3	1, 2	mpbir 200	1 ⊢ ∅ ∈ 0c

Syntax hints:   = wceq 1642   ∈ wcel 1710  ∅c0 3551  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:  ncfinraise  4482  ncfinlower  4484  tfin0c  4498  nulnnn  4557  tc0c  6164  ce0nnul  6178  ce0nn  6181  ce0nulnc  6185