Theorem snopeqopsnid 5417

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

Step	Hyp	Ref	Expression
1		eqid 2738	. 2 ⊢ 𝐴 = 𝐴
2		eqid 2738	. 2 ⊢ {𝐴} = {𝐴}
3		snopeqopsnid.a	. . 3 ⊢ 𝐴 ∈ V
4	3, 3	snopeqop 5414	. 2 ⊢ ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
5	1, 2, 2, 4	mpbir3an 1339	1 ⊢ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩

Syntax hints:   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558  ⟨cop 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565

This theorem is referenced by:  funsneqopb  7006  vtxvalsnop  27314  iedgvalsnop  27315