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Theorem exdifsn 32372
 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}) ↔ ∃𝑥𝐴 𝑥𝐵)

Proof of Theorem exdifsn
StepHypRef Expression
1 eldifsn 4704 . . 3 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
21exbii 1849 . 2 (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
3 df-rex 3139 . 2 (∃𝑥𝐴 𝑥𝐵 ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
42, 3bitr4i 281 1 (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥𝐴 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781   ∈ wcel 2115   ≠ wne 3014  ∃wrex 3134   ∖ cdif 3916  {csn 4550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3015  df-rex 3139  df-v 3482  df-dif 3922  df-sn 4551 This theorem is referenced by: (None)
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