Theorem rex0 4301

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 4279	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3066	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2114  ∃wrex 3062  ∅c0 4274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-dif 3893  df-nul 4275

This theorem is referenced by:  reu0  4302  rmo0  4303  0iun  5006  0qs  8702  sup0riota  9372  cfeq0  10169  cfsuc  10170  hashge2el2difr  14434  cshws0  17063  addsrid  27970  muls01  28118  mulsrid  28119  elons2  28264  onaddscl  28283  onmulscl  28284  n0cut  28340  0ringirng  33849  dya2iocuni  34443  eulerpartlemgh  34538  pmapglb2xN  40232  elpadd0  40269  tfsconcatb0  43790  sprsymrelfvlem  47962  ipolub00  49480