Theorem brtpid3 33544

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

Assertion

Ref	Expression
brtpid3	⊢ 𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵

Proof of Theorem brtpid3

Step	Hyp	Ref	Expression
1		opex 5372	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid3 4706	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {𝐶, 𝐷, ⟨𝐴, 𝐵⟩}
3		df-br 5071	. 2 ⊢ (𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {𝐶, 𝐷, ⟨𝐴, 𝐵⟩})
4	2, 3	mpbir 234	1 ⊢ 𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵

Syntax hints:   ∈ wcel 2112  {ctp 4562  ⟨cop 4564   class class class wbr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710  ax-sep 5216  ax-nul 5223  ax-pr 5346

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-v 3425  df-dif 3887  df-un 3889  df-nul 4255  df-if 4457  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-br 5071

This theorem is referenced by: (None)