Theorem rex0 4311

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 4286	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 120	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3094	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2147  ∃wrex 3090  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-dif 4050  df-nul 4282

This theorem is used by:  reu0  4312  rmo0  4313  rab0  4338  0iun  5024  0qs  8811  sup0riota  9485  cfeq0  10361  cfsuc  10362  hashge2el2difr  14653  cshws0  17308  addsrid  28380  muls01  28528  mulsrid  28529  elons2  28674  onaddscl  28693  onmulscl  28694  n0cut  28750  0ringirng  34391  dya2iocuni  34985  eulerpartlemgh  35080  pmapglb2xN  41021  elpadd0  41058  tfsconcatb0  44538  sprsymrelfvlem  48777  ipolub00  50312