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

Theorem snopeqopsnid 5514
Description: Equivalence for an ordered pair of two identical singletons equal to a singleton of an ordered pair. (Contributed by AV, 24-Sep-2020.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.)
Hypothesis
Ref Expression
snopeqopsnid.a 𝐴 ∈ V
Assertion
Ref Expression
snopeqopsnid {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩

Proof of Theorem snopeqopsnid
StepHypRef Expression
1 eqid 2737 . 2 𝐴 = 𝐴
2 eqid 2737 . 2 {𝐴} = {𝐴}
3 snopeqopsnid.a . . 3 𝐴 ∈ V
43, 3snopeqop 5511 . 2 ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
51, 2, 2, 4mpbir3an 1342 1 {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632
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  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633
This theorem is referenced by:  funsneqopb  7172  vtxvalsnop  29058  iedgvalsnop  29059
  Copyright terms: Public domain W3C validator