Theorem exdifsn 35091

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 4735	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2	1	exbii 1849	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
3		df-rex 3057	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ≠ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
4	2, 3	bitr4i 278	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056   ∖ cdif 3894  {csn 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rex 3057  df-v 3438  df-dif 3900  df-sn 4574

This theorem is referenced by: (None)