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Theorem mosneq 4842
Description: There exists at most one set whose singleton is equal to a given class. See also moeq 3713. (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 2763 . . . 4 (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦})
2 vex 3484 . . . . 5 𝑥 ∈ V
32sneqr 4840 . . . 4 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
41, 3syl 17 . . 3 (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
54gen2 1796 . 2 𝑥𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
6 sneq 4636 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
76eqeq1d 2739 . . 3 (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴))
87mo4 2566 . 2 (∃*𝑥{𝑥} = 𝐴 ↔ ∀𝑥𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦))
95, 8mpbir 231 1 ∃*𝑥{𝑥} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538   = wceq 1540  ∃*wmo 2538  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  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 1543  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sn 4627
This theorem is referenced by:  pwfir  9355  euabsneu  47040
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