Theorem tpnz 3798

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 3783	. 2 |- A e. { A , B , C }
3		ne0i 3501	. 2 |- ( A e. { A , B , C } -> { A , B , C } =/= (/) )
4	2, 3	ax-mp 5	1 |- { A , B , C } =/= (/)

Syntax hints:   e. wcel 2202   =/= wne 2402   _Vcvv 2802   (/)c0 3494   {ctp 3671

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-un 3204  df-nul 3495  df-sn 3675  df-pr 3676  df-tp 3677

This theorem is referenced by: (None)