Theorem brtpid1 36250

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 5447	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid1 4736	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩, 𝐶, 𝐷}
3		df-br 5112	. 2 ⊢ (𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩, 𝐶, 𝐷})
4	2, 3	mpbir 234	1 ⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵

This proof depends on syntax axioms:   ∈ wcel 2146  {ctp 4595  ⟨cop 4597   class class class wbr 5111

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-un 3911  df-in 3913  df-ss 3923  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-br 5112

This theorem is used by: (None)