Theorem brtpid3 32948

Description: A binary relation involving unordered triplets. (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 5348	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	tpid3 4702	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {𝐶, 𝐷, ⟨𝐴, 𝐵⟩}
3		df-br 5059	. 2 ⊢ (𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {𝐶, 𝐷, ⟨𝐴, 𝐵⟩})
4	2, 3	mpbir 233	1 ⊢ 𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵

Syntax hints:   ∈ wcel 2110  {ctp 4564  ⟨cop 4566   class class class wbr 5058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-br 5059

This theorem is referenced by: (None)