Theorem brtpid1 36108

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

Assertion

Ref	Expression
brtpid1	⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵

Proof of Theorem brtpid1

Step	Hyp	Ref	Expression
1		opex 5443	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid1 4736	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩, 𝐶, 𝐷}
3		df-br 5111	. 2 ⊢ (𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩, 𝐶, 𝐷})
4	2, 3	mpbir 234	1 ⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵

Syntax hints:   ∈ wcel 2149  {ctp 4595  ⟨cop 4597   class class class wbr 5110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-un 3918  df-in 3920  df-ss 3930  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-br 5111

This theorem is referenced by: (None)