Theorem brtpid3 33066

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 5321	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid3 4669	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {𝐶, 𝐷, ⟨𝐴, 𝐵⟩}
3		df-br 5031	. 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 2111  {ctp 4529  ⟨cop 4531   class class class wbr 5030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-br 5031

This theorem is referenced by: (None)