Theorem exdifsn 35276

Description: There exists an element in a class excluding a singleton if and only if there exists an element in the original class not equal to the singleton element. (Contributed by BTernaryTau, 15-Sep-2023.)

Assertion

Ref	Expression
exdifsn	⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 𝐵)

Proof of Theorem exdifsn

Step	Hyp	Ref	Expression
1		eldifsn 4722	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2	1	exbii 1856	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
3		df-rex 3066	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ≠ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
4	2, 3	bitr4i 280	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 𝐵)

Syntax hints:   ↔ wb 208   ∧ wa 397  ∃wex 1787   ∈ wcel 2121   ≠ wne 2936  ∃wrex 3065   ∖ cdif 3882  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-rex 3066  df-v 3435  df-dif 3888  df-sn 4559

This theorem is referenced by: (None)