Theorem tpnz 3796

Description: A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)

Hypothesis

Ref	Expression
tpnz.1	|- A e. _V

Assertion

Ref	Expression
tpnz	|- { A , B , C } =/= (/)

Proof of Theorem tpnz

Step	Hyp	Ref	Expression
1		tpnz.1	. . 3 |- A e. _V
2	1	tpid1 3781	. 2 |- A e. { A , B , C }
3		ne0i 3499	. 2 |- ( A e. { A , B , C } -> { A , B , C } =/= (/) )
4	2, 3	ax-mp 5	1 |- { A , B , C } =/= (/)

Syntax hints:   e. wcel 2200   =/= wne 2400   _Vcvv 2800   (/)c0 3492   {ctp 3669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2802  df-dif 3200  df-un 3202  df-nul 3493  df-sn 3673  df-pr 3674  df-tp 3675

This theorem is referenced by: (None)