Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpnz Structured version   Visualization version   GIF version

Theorem tpnz 4457
 Description: A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
Hypothesis
Ref Expression
tpnz.1 𝐴 ∈ V
Assertion
Ref Expression
tpnz {𝐴, 𝐵, 𝐶} ≠ ∅

Proof of Theorem tpnz
StepHypRef Expression
1 tpnz.1 . . 3 𝐴 ∈ V
21tpid1 4448 . 2 𝐴 ∈ {𝐴, 𝐵, 𝐶}
32ne0ii 4067 1 {𝐴, 𝐵, 𝐶} ≠ ∅
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2140   ≠ wne 2933  Vcvv 3341  ∅c0 4059  {ctp 4326 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-v 3343  df-dif 3719  df-un 3721  df-nul 4060  df-sn 4323  df-pr 4325  df-tp 4327 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator