Theorem opid 4359

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.) (Avoid depending on this detail.)

Hypothesis

Ref	Expression
opid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
opid	⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Proof of Theorem opid

Step	Hyp	Ref	Expression
1		dfsn2 4138	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	preq2i 4216	. 2 ⊢ {{𝐴}, {𝐴}} = {{𝐴}, {𝐴, 𝐴}}
3		dfsn2 4138	. 2 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
4		opid.1	. . 3 ⊢ 𝐴 ∈ V
5	4, 4	dfop 4339	. 2 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
6	2, 3, 5	3eqtr4ri 2643	1 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  {cpr 4127  ⟨cop 4131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132

This theorem is referenced by:  dmsnsnsn  5531  funopg  5836  vtxval3sn  25718  iedgval3sn  25719