Theorem rex0 4316

Description: Vacuous restricted 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 4293	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 120	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3093	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2145  ∃wrex 3089  ∅c0 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-dif 3910  df-nul 4289

This theorem is referenced by:  reu0  4317  rmo0  4318  rab0  4342  0iun  5023  0qs  8748  sup0riota  9414  cfeq0  10228  cfsuc  10229  hashge2el2difr  14508  cshws0  17151  addsrid  28115  muls01  28263  mulsrid  28264  elons2  28409  onaddscl  28428  onmulscl  28429  n0cut  28485  0ringirng  33996  dya2iocuni  34590  eulerpartlemgh  34685  pmapglb2xN  40408  elpadd0  40445  tfsconcatb0  43933  sprsymrelfvlem  48094  ipolub00  49622