Theorem snopeqopsnid 5472

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 2730	. 2 ⊢ 𝐴 = 𝐴
2		eqid 2730	. 2 ⊢ {𝐴} = {𝐴}
3		snopeqopsnid.a	. . 3 ⊢ 𝐴 ∈ V
4	3, 3	snopeqop 5469	. 2 ⊢ ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
5	1, 2, 2, 4	mpbir3an 1342	1 ⊢ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {csn 4592  ⟨cop 4598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599

This theorem is referenced by:  funsneqopb  7127  vtxvalsnop  28975  iedgvalsnop  28976