Theorem brtpid1 35348

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 5470	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid1 4777	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩, 𝐶, 𝐷}
3		df-br 5153	. 2 ⊢ (𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩, 𝐶, 𝐷})
4	2, 3	mpbir 230	1 ⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵

Syntax hints:   ∈ wcel 2098  {ctp 4636  ⟨cop 4638   class class class wbr 5152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-br 5153

This theorem is referenced by: (None)