Theorem 0pss 3852

Description: The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.)

Assertion

Ref	Expression
0pss	|- ( (/) C. A <-> A =/= (/) )

Proof of Theorem 0pss

Step	Hyp	Ref	Expression
1		0ss 3813	. . 3 |- (/) C_ A
2		df-pss 3477	. . 3 |- ( (/) C. A <-> ( (/) C_ A /\ (/) =/= A ) )
3	1, 2	mpbiran 916	. 2 |- ( (/) C. A <-> (/) =/= A )
4		necom 2723	. 2 |- ( (/) =/= A <-> A =/= (/) )
5	3, 4	bitri 249	1 |- ( (/) C. A <-> A =/= (/) )

Syntax hints:   <-> wb 184   =/= wne 2649   C_ wss 3461   C. wpss 3462   (/)c0 3783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784

This theorem is referenced by:  php  7694  zornn0g  8876  prn0  9356  genpn0  9370  nqpr  9381  ltexprlem5  9407  reclem2pr  9415  suplem1pr  9419  alexsubALTlem4  20716  bj-2upln0  34982  bj-2upln1upl  34983  0pssin  38244