Theorem rexn0 4505

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2803, ax-8 2101. (Revised by GG, 2-Sep-2024.)

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		dfrex2 3063	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		rzal 4503	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ¬ 𝜑)
3	2	con3i 154	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ 𝐴 = ∅)
4	1, 3	sylbi 216	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ¬ 𝐴 = ∅)
5	4	neqned 2937	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  ∅c0 4322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-ne 2931  df-ral 3052  df-rex 3061  df-dif 3949  df-nul 4323

This theorem is referenced by:  2reu4  4521  reusv2lem3  5396  eusvobj2  7408  isdrs2  18326  ismnd  18725  slwn0  19609  lbsexg  21141  iunconn  23420  sltn0  27925  grpon0  30432  filbcmb  37454  isbnd2  37497  rencldnfi  42515  iunconnlem2  44648  stoweidlem14  45671  hoidmvval0  46244  thinciso  48417