Theorem rexn0 4441

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2816, ax-8 2108. (Revised by Gino Giotto, 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 3170	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		rzal 4439	. . . 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 2950	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-ne 2944  df-ral 3069  df-rex 3070  df-dif 3890  df-nul 4257

This theorem is referenced by:  2reu4  4457  reusv2lem3  5323  eusvobj2  7268  isdrs2  18024  ismnd  18388  slwn0  19220  lbsexg  20426  iunconn  22579  grpon0  28864  sltn0  34085  filbcmb  35898  isbnd2  35941  rencldnfi  40643  iunconnlem2  42555  stoweidlem14  43555  hoidmvval0  44125  thinciso  46341