Theorem opid 4893

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Inference form of opidg 4892. (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 4892	. 2 ⊢ (𝐴 ∈ V → ⟨𝐴, 𝐴⟩ = {{𝐴}})
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Syntax hints:   = wceq 1540   ∈ wcel 2108  Vcvv 3480  {csn 4626  ⟨cop 4632

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633

This theorem is referenced by:  dmsnsnsn  6240  funopg  6600  vtxval3sn  29060  iedgval3sn  29061