Theorem rex0 3564

Description: Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
rex0	⊢ ¬ ∃x ∈ ∅ φ

Proof of Theorem rex0

Step	Hyp	Ref	Expression
1		noel 3555	. . 3 ⊢ ¬ x ∈ ∅
2	1	pm2.21i 123	. 2 ⊢ (x ∈ ∅ → ¬ φ)
3	2	nrex 2717	1 ⊢ ¬ ∃x ∈ ∅ φ

Syntax hints:  ¬ wn 3   ∈ wcel 1710  ∃wrex 2616  ∅c0 3551

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

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-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  0iun  4024  addcnul1  4453  nnadjoin  4521  tfinnn  4535