Theorem rexn0 4026

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Assertion

Ref	Expression
rexn0	⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexn0

Step	Hyp	Ref	Expression
1		ne0i 3880	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	a1d 25	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝐴 ≠ ∅))
3	2	rexlimiv 3009	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 1977   ≠ wne 2780  ∃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-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by:  reusv2lem3  4797  eusvobj2  6542  isdrs2  16762  ismnd  17120  slwn0  17853  lbsexg  18985  iuncon  21041  grpon0  26740  filbcmb  32705  isbnd2  32752  rencldnfi  36403  iunconlem2  38193  stoweidlem14  38907  hoidmvval0  39477  2reu4  39839