Theorem brtpid3 36409

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 5431	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid3 4733	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {𝐶, 𝐷, ⟨𝐴, 𝐵⟩}
3		df-br 5103	. 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

This proof depends on syntax axioms:   ∈ wcel 2145  {ctp 4587  ⟨cop 4589   class class class wbr 5102

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-un 3903  df-in 3905  df-ss 3915  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-br 5103

This theorem is used by: (None)