Theorem brtpid2 35714

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 5476	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid2 4776	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {𝐶, ⟨𝐴, 𝐵⟩, 𝐷}
3		df-br 5150	. 2 ⊢ (𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {𝐶, ⟨𝐴, 𝐵⟩, 𝐷})
4	2, 3	mpbir 231	1 ⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵

Syntax hints:   ∈ wcel 2107  {ctp 4636  ⟨cop 4638   class class class wbr 5149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-br 5150

This theorem is referenced by: (None)