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Theorem snopeqopsnid 5519
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 2735 . 2 𝐴 = 𝐴
2 eqid 2735 . 2 {𝐴} = {𝐴}
3 snopeqopsnid.a . . 3 𝐴 ∈ V
43, 3snopeqop 5516 . 2 ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
51, 2, 2, 4mpbir3an 1340 1 {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631  cop 4637
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  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  funsneqopb  7172  vtxvalsnop  29073  iedgvalsnop  29074
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