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Theorem tpnzd 4119
Description: A triplet containing a set is not empty. (Contributed by Thierry Arnoux, 8-Apr-2019.)
Hypothesis
Ref Expression
tpnzd.1  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
tpnzd  |-  ( ph  ->  { A ,  B ,  C }  =/=  (/) )

Proof of Theorem tpnzd
StepHypRef Expression
1 tpnzd.1 . 2  |-  ( ph  ->  A  e.  V )
2 tpid3g 4112 . . 3  |-  ( A  e.  V  ->  A  e.  { B ,  C ,  A } )
3 tprot 4092 . . 3  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
42, 3syl6eleqr 2521 . 2  |-  ( A  e.  V  ->  A  e.  { A ,  B ,  C } )
5 ne0i 3767 . 2  |-  ( A  e.  { A ,  B ,  C }  ->  { A ,  B ,  C }  =/=  (/) )
61, 4, 53syl 18 1  |-  ( ph  ->  { A ,  B ,  C }  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1868    =/= wne 2618   (/)c0 3761   {ctp 4000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-v 3083  df-dif 3439  df-un 3441  df-nul 3762  df-sn 3997  df-pr 3999  df-tp 4001
This theorem is referenced by:  raltpd  4120  fr3nr  6617  etransclem48OLD  37967  etransclem48  37968
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