Theorem tpid1 3777

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
tpid1.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tpid1	⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid1

Step	Hyp	Ref	Expression
1		eqid 2229	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1193	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)
3		tpid1.1	. . 3 ⊢ 𝐴 ∈ V
4	3	eltp 3714	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	2, 4	mpbir 146	1 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 1001   = wceq 1395   ∈ wcel 2200  Vcvv 2799  {ctp 3668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-tp 3674

This theorem is referenced by:  tpnz  3792