Theorem brtpid2 34679

Description: A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)

Assertion

Ref	Expression
brtpid2	⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵

Proof of Theorem brtpid2

Step	Hyp	Ref	Expression
1		opex 5463	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid2 4773	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {𝐶, ⟨𝐴, 𝐵⟩, 𝐷}
3		df-br 5148	. 2 ⊢ (𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {𝐶, ⟨𝐴, 𝐵⟩, 𝐷})
4	2, 3	mpbir 230	1 ⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵

Syntax hints:   ∈ wcel 2106  {ctp 4631  ⟨cop 4633   class class class wbr 5147

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-br 5148

This theorem is referenced by: (None)