Theorem opid 4851

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Inference form of opidg 4850. (Contributed by FL, 28-Dec-2011.) (Proof shortened by AV, 16-Feb-2022.) (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		opid.1	. 2 ⊢ 𝐴 ∈ V
2		opidg 4850	. 2 ⊢ (𝐴 ∈ V → ⟨𝐴, 𝐴⟩ = {{𝐴}})
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582  ⟨cop 4588

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by:  dmsnsnsn  6186  funopg  6534  vtxval3sn  29128  iedgval3sn  29129