Theorem rexn0 4464

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2803, ax-8 2111. (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 3056	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		rzal 4462	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ¬ 𝜑)
3	2	con3i 154	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ 𝐴 = ∅)
4	1, 3	sylbi 217	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ¬ 𝐴 = ∅)
5	4	neqned 2932	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-ne 2926  df-ral 3045  df-rex 3054  df-dif 3908  df-nul 4287

This theorem is referenced by:  2reu4  4476  reusv2lem3  5342  eusvobj2  7345  isdrs2  18230  ismnd  18629  slwn0  19512  lbsexg  21089  iunconn  23331  sltn0  27838  grpon0  30464  filbcmb  37719  isbnd2  37762  rencldnfi  42794  iunconnlem2  44908  stoweidlem14  45996  hoidmvval0  46569  thinciso  49456