Theorem exdifsn 35057

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 4811	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2	1	exbii 1846	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
3		df-rex 3077	. 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 1777   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076   ∖ cdif 3973  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rex 3077  df-v 3490  df-dif 3979  df-sn 4649

This theorem is referenced by: (None)