Theorem exdifsn 35437

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 4758	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2	1	exbii 1876	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
3		df-rex 3097	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ≠ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
4	2, 3	bitr4i 281	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 𝐵)

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1807   ∈ wcel 2150   ≠ wne 2965  ∃wrex 3096   ∖ cdif 3910  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ne 2966  df-rex 3097  df-v 3464  df-dif 3916  df-sn 4595

This theorem is referenced by: (None)