Theorem rexn0 4454

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 4300	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	a1d 25	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝐴 ≠ ∅))
3	2	rexlimiv 3280	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 3016  ∃wrex 3139  ∅c0 4291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-ral 3143  df-rex 3144  df-dif 3939  df-nul 4292

This theorem is referenced by:  2reu4  4466  reusv2lem3  5301  eusvobj2  7149  isdrs2  17549  ismnd  17914  slwn0  18740  lbsexg  19936  iunconn  22036  grpon0  28279  filbcmb  35030  isbnd2  35076  rencldnfi  39438  iunconnlem2  41289  stoweidlem14  42319  hoidmvval0  42889