Theorem tpid2 4247

Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypothesis

Ref	Expression
tpid2.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
tpid2	⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid2

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ 𝐵 = 𝐵
2	1	3mix2i 1227	. 2 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)
3		tpid2.1	. . 3 ⊢ 𝐵 ∈ V
4	3	eltp 4177	. 2 ⊢ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶))
5	2, 4	mpbir 220	1 ⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {ctp 4129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-tp 4130

This theorem is referenced by:  wrdl3s3  13553  2pthlem1  26125  2pthlem2  26126  el2wlkonotlem  26389  sgnsf  29060  sgncl  29927  signsw0glem  29956  signsw0g  29959  signswmnd  29960  signswrid  29961  kur14lem7  30448  brtpid2  30858  rabren3dioph  36397  fourierdlem102  39101  fourierdlem114  39113  etransclem48  39175  wwlks2onv  41158  elwwlks2ons3  41159  umgrwwlks2on  41161