Theorem rexn0 4437

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2812, ax-8 2116. (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 3065	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		rzal 4435	. . . 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 2940	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  ∅c0 4274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-ne 2934  df-ral 3053  df-rex 3063  df-dif 3893  df-nul 4275

This theorem is referenced by:  r19.2zb  4441  2reu4  4465  reusv2lem3  5343  eusvobj2  7359  isdrs2  18272  ismnd  18705  slwn0  19590  lbsexg  21162  iunconn  23393  ltsn0  27898  grpon0  30573  filbcmb  38061  isbnd2  38104  rencldnfi  43249  iunconnlem2  45361  stoweidlem14  46442  hoidmvval0  47015  thinciso  49939