Theorem noel 3554

Description: The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Assertion

Ref	Expression
noel	⊢ ¬ A ∈ ∅

Proof of Theorem noel

Step	Hyp	Ref	Expression
1		eldifi 3388	. . 3 ⊢ (A ∈ (V ∖ V) → A ∈ V)
2		eldifn 3389	. . 3 ⊢ (A ∈ (V ∖ V) → ¬ A ∈ V)
3	1, 2	pm2.65i 165	. 2 ⊢ ¬ A ∈ (V ∖ V)
4		df-nul 3551	. . 3 ⊢ ∅ = (V ∖ V)
5	4	eleq2i 2417	. 2 ⊢ (A ∈ ∅ ↔ A ∈ (V ∖ V))
6	3, 5	mtbir 290	1 ⊢ ¬ A ∈ ∅

Syntax hints:  ¬ wn 3   ∈ wcel 1710  Vcvv 2859   ∖ cdif 3206  ∅c0 3550

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

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-dif 3215  df-nul 3551

This theorem is referenced by:  n0i  3555  n0f  3558  rex0  3563  abvor0  3567  rab0  3571  un0  3575  in0  3576  0ss  3579  r19.2zb  3640  ral0  3654  int0  3940  iun0  4022  0iun  4023  vinf  4555  xp0r  4842  dm0  4918  dm0rn0  4921  dmeq0  4922  cnv0  5031  co02  5092  co01  5093  fv01  5354  clos10  5887  po0  5939  connex0  5940