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

Step	Hyp	Ref	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 }
4	2, 3	syl6eleqr 2542	. 2 |- ( A e. V -> A e. { A , B , C } )
5		ne0i 3739	. 2 |- ( A e. { A , B , C } -> { A , B , C } =/= (/) )
6	1, 4, 5	3syl 18	1 |- ( ph -> { A , B , C } =/= (/) )

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