Theorem tpnzd 4314

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 4305	. . 3 |- ( A e. V -> A e. { B , C , A } )
3		tprot 4284	. . 3 |- { A , B , C } = { B , C , A }
4	2, 3	syl6eleqr 2712	. 2 |- ( A e. V -> A e. { A , B , C } )
5		ne0i 3921	. 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 1990   =/= wne 2794   (/)c0 3915   {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  raltpd  4315  fr3nr  6979  limsupequzlem  39954  etransclem48  40499