Theorem exdifsn 35086

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 4794	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2	1	exbii 1847	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
3		df-rex 3071	. 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 1778   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070   ∖ cdif 3963  {csn 4634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rex 3071  df-v 3483  df-dif 3969  df-sn 4635

This theorem is referenced by: (None)