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Theorem snopeqopsnid 5465
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 5462 . 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 1541  wcel 2106  Vcvv 3444  {csn 4585  cop 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2943  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592
This theorem is referenced by:  funsneqopb  7095  vtxvalsnop  27890  iedgvalsnop  27891
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