MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mosneq Structured version   Visualization version   GIF version

Theorem mosneq 4847
Description: There exists at most one set whose singleton is equal to a given class. See also moeq 3716. (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 2761 . . . 4 (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦})
2 vex 3482 . . . . 5 𝑥 ∈ V
32sneqr 4845 . . . 4 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
41, 3syl 17 . . 3 (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
54gen2 1793 . 2 𝑥𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
6 sneq 4641 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
76eqeq1d 2737 . . 3 (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴))
87mo4 2564 . 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 1535   = wceq 1537  ∃*wmo 2536  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sn 4632
This theorem is referenced by:  pwfir  9353  euabsneu  46978
  Copyright terms: Public domain W3C validator