Theorem brtpid2 36172

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

Assertion

Ref	Expression
brtpid2	⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵

Proof of Theorem brtpid2

Step	Hyp	Ref	Expression
1		opex 5449	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid2 4741	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {𝐶, ⟨𝐴, 𝐵⟩, 𝐷}
3		df-br 5115	. 2 ⊢ (𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {𝐶, ⟨𝐴, 𝐵⟩, 𝐷})
4	2, 3	mpbir 234	1 ⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵

Syntax hints:   ∈ wcel 2150  {ctp 4598  ⟨cop 4600   class class class wbr 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-un 3918  df-in 3920  df-ss 3930  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-br 5115

This theorem is referenced by: (None)