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

Step	Hyp	Ref	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 }
4	2, 3	syl6eleqr 2521	. 2 |- ( A e. V -> A e. { A , B , C } )
5		ne0i 3767	. 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 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