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Theorem mosneq 4756
Description: There exists at most one set whose singleton is equal to a given class. See also moeq 3683. (Contributed by BJ, 24-Sep-2022.)
Assertion
Ref Expression
mosneq ∃*𝑥{𝑥} = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem mosneq
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2846 . . . 4 (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦})
2 vex 3482 . . . . 5 𝑥 ∈ V
32sneqr 4754 . . . 4 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
41, 3syl 17 . . 3 (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
54gen2 1798 . 2 𝑥𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
6 sneq 4558 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
76eqeq1d 2826 . . 3 (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴))
87mo4 2651 . 2 (∃*𝑥{𝑥} = 𝐴 ↔ ∀𝑥𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦))
95, 8mpbir 234 1 ∃*𝑥{𝑥} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  ∃*wmo 2622  {csn 4548
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-mo 2624  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3481  df-sn 4549
This theorem is referenced by:  euabsneu  43477
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