Theorem nulel0c 4422

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 4421	. 2 ⊢ (∅ ∈ 0c ↔ ∅ = ∅)
3	1, 2	mpbir 200	1 ⊢ ∅ ∈ 0c

Syntax hints:   = wceq 1642   ∈ wcel 1710  ∅c0 3550  0_cc0c 4374

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 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:  ncfinraise  4481  ncfinlower  4483  tfin0c  4497  nulnnn  4556  tc0c  6163  ce0nnul  6177  ce0nn  6180  ce0nulnc  6184