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Theorem tpnzd 4097
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 4090 . . 3  |-  ( A  e.  V  ->  A  e.  { B ,  C ,  A } )
3 tprot 4070 . . 3  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
42, 3syl6eleqr 2542 . 2  |-  ( A  e.  V  ->  A  e.  { A ,  B ,  C } )
5 ne0i 3739 . 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 1889    =/= wne 2624   (/)c0 3733   {ctp 3974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-v 3049  df-dif 3409  df-un 3411  df-nul 3734  df-sn 3971  df-pr 3973  df-tp 3975
This theorem is referenced by:  raltpd  4098  fr3nr  6611  etransclem48OLD  38157  etransclem48  38158
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