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

Theorem snopeqopsnid 5432
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 2736 . 2 𝐴 = 𝐴
2 eqid 2736 . 2 {𝐴} = {𝐴}
3 snopeqopsnid.a . . 3 𝐴 ∈ V
43, 3snopeqop 5429 . 2 ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
51, 2, 2, 4mpbir3an 1341 1 {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2104  Vcvv 3437  {csn 4565  cop 4571
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572
This theorem is referenced by:  funsneqopb  7052  vtxvalsnop  27452  iedgvalsnop  27453
  Copyright terms: Public domain W3C validator