Theorem rexn0 4453

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2840, ax-8 2147. (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 3092	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		rzal 4451	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ¬ 𝜑)
3	2	con3i 155	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ 𝐴 = ∅)
4	1, 3	sylbi 220	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ¬ 𝐴 = ∅)
5	4	neqned 2967	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1563   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  ∅c0 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-ne 2961  df-ral 3080  df-rex 3090  df-dif 3910  df-nul 4289

This theorem is referenced by:  r19.2zb  4457  2reu4  4481  reusv2lem3  5362  eusvobj2  7392  isdrs2  18352  ismnd  18785  slwn0  19676  lbsexg  21257  iunconn  23546  ltsn0  28057  grpon0  30763  filbcmb  38251  isbnd2  38294  rencldnfi  43410  iunconnlem2  45508  stoweidlem14  46586  hoidmvval0  47159  thinciso  50099