Theorem snopeqopsnid 5485

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 2763	. 2 ⊢ 𝐴 = 𝐴
2		eqid 2763	. 2 ⊢ {𝐴} = {𝐴}
3		snopeqopsnid.a	. . 3 ⊢ 𝐴 ∈ V
4	3, 3	snopeqop 5482	. 2 ⊢ ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
5	1, 2, 2, 4	mpbir3an 1360	1 ⊢ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {csn 4588  ⟨cop 4838

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839

This theorem is used by:  funsneqopb  7180  vtxvalsnop  29649  iedgvalsnop  29650