Theorem snopeqopsnid 5368

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 2758	. 2 ⊢ 𝐴 = 𝐴
2		eqid 2758	. 2 ⊢ {𝐴} = {𝐴}
3		snopeqopsnid.a	. . 3 ⊢ 𝐴 ∈ V
4	3, 3	snopeqop 5365	. 2 ⊢ ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
5	1, 2, 2, 4	mpbir3an 1338	1 ⊢ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩

Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3409  {csn 4522  ⟨cop 4528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-v 3411  df-dif 3861  df-un 3863  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529

This theorem is referenced by:  funsneqopb  6905  vtxvalsnop  26933  iedgvalsnop  26934