Theorem rex0 3894

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 3878	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 115	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 2983	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 1977  ∃wrex 2897  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by:  0iun  4513  sup0riota  8254  cfeq0  8961  cfsuc  8962  hashge2el2difr  13118  cshws0  15646  meet0  16960  join0  16961  dya2iocuni  29672  eulerpartlemgh  29767  pmapglb2xN  34076  elpadd0  34113