Theorem rex0 4312

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 4290	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3089	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2141  ∃wrex 3085  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-dif 3907  df-nul 4286

This theorem is referenced by:  reu0  4313  rmo0  4314  rab0  4338  0iun  5019  0qs  8739  sup0riota  9409  cfeq0  10210  cfsuc  10211  hashge2el2difr  14491  cshws0  17120  addsrid  28034  muls01  28182  mulsrid  28183  elons2  28328  onaddscl  28347  onmulscl  28348  n0cut  28404  0ringirng  33947  dya2iocuni  34541  eulerpartlemgh  34636  pmapglb2xN  40360  elpadd0  40397  tfsconcatb0  43885  sprsymrelfvlem  48060  ipolub00  49578