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Theorem mosneq 4773
Description: There exists at most one set whose singleton is equal to a given class. See also moeq 3642. (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 2764 . . . 4 (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦})
2 vex 3436 . . . . 5 𝑥 ∈ V
32sneqr 4771 . . . 4 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
41, 3syl 17 . . 3 (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
54gen2 1799 . 2 𝑥𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
6 sneq 4571 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
76eqeq1d 2740 . . 3 (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴))
87mo4 2566 . 2 (∃*𝑥{𝑥} = 𝐴 ↔ ∀𝑥𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦))
95, 8mpbir 230 1 ∃*𝑥{𝑥} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1537   = wceq 1539  ∃*wmo 2538  {csn 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-sn 4562
This theorem is referenced by:  pwfir  8959  euabsneu  44522
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