Theorem snopeqopsnid 5492

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 2761	. 2 ⊢ 𝐴 = 𝐴
2		eqid 2761	. 2 ⊢ {𝐴} = {𝐴}
3		snopeqopsnid.a	. . 3 ⊢ 𝐴 ∈ V
4	3, 3	snopeqop 5489	. 2 ⊢ ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
5	1, 2, 2, 4	mpbir3an 1358	1 ⊢ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩

Syntax hints:   = wceq 1568   ∈ wcel 2141  Vcvv 3453  {csn 4588  ⟨cop 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595

This theorem is referenced by:  funsneqopb  7149  vtxvalsnop  29357  iedgvalsnop  29358