Theorem tpid2 4709

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 2824	. . 3 ⊢ 𝐵 = 𝐵
2	1	3mix2i 1330	. 2 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)
3		tpid2.1	. . 3 ⊢ 𝐵 ∈ V
4	3	eltp 4629	. 2 ⊢ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶))
5	2, 4	mpbir 233	1 ⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 1082   = wceq 1536   ∈ wcel 2113  Vcvv 3497  {ctp 4574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-un 3944  df-sn 4571  df-pr 4573  df-tp 4575

This theorem is referenced by:  wrdl3s3  14329  wwlks2onv  27735  elwwlks2ons3im  27736  umgrwwlks2on  27739  s3rn  30626  cyc3evpm  30796  sgnsf  30808  sgncl  31800  signsw0glem  31827  signsw0g  31830  signswmnd  31831  signswrid  31832  prodfzo03  31878  circlevma  31917  circlemethhgt  31918  hgt750lemg  31929  hgt750lemb  31931  hgt750lema  31932  hgt750leme  31933  tgoldbachgtde  31935  tgoldbachgt  31938  kur14lem7  32463  brtpid2  32956  rabren3dioph  39418  fourierdlem102  42500  fourierdlem114  42512  etransclem48  42574