Theorem brtpid1 35705

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 5432	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid1 4740	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩, 𝐶, 𝐷}
3		df-br 5116	. 2 ⊢ (𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩, 𝐶, 𝐷})
4	2, 3	mpbir 231	1 ⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵

Syntax hints:   ∈ wcel 2109  {ctp 4601  ⟨cop 4603   class class class wbr 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-tp 4602  df-op 4604  df-br 5116

This theorem is referenced by: (None)