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Theorem mosneq 4779
Description: There exists at most one set whose singleton is equal to a given class. See also moeq 3646. (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 2766 . . . 4 (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦})
2 vex 3435 . . . . 5 𝑥 ∈ V
32sneqr 4777 . . . 4 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
41, 3syl 17 . . 3 (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
54gen2 1803 . 2 𝑥𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
6 sneq 4577 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
76eqeq1d 2742 . . 3 (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴))
87mo4 2568 . 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 1540   = wceq 1542  ∃*wmo 2540  {csn 4567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-mo 2542  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-sn 4568
This theorem is referenced by:  pwfir  8939  euabsneu  44488
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