Theorem brtpid1 35949

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 5403	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid1 4700	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩, 𝐶, 𝐷}
3		df-br 5073	. 2 ⊢ (𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩, 𝐶, 𝐷})
4	2, 3	mpbir 232	1 ⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵

Syntax hints:   ∈ wcel 2119  {ctp 4559  ⟨cop 4561   class class class wbr 5072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-un 3888  df-in 3890  df-ss 3900  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-br 5073

This theorem is referenced by: (None)