Theorem 0pss 3827

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 3777	. . 3 |- (/) C_ A
2		df-pss 3455	. . 3 |- ( (/) C. A <-> ( (/) C_ A /\ (/) =/= A ) )
3	1, 2	mpbiran 909	. 2 |- ( (/) C. A <-> (/) =/= A )
4		necom 2721	. 2 |- ( (/) =/= A <-> A =/= (/) )
5	3, 4	bitri 249	1 |- ( (/) C. A <-> A =/= (/) )

Syntax hints:   <-> wb 184   =/= wne 2648   C_ wss 3439   C. wpss 3440   (/)c0 3748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-dif 3442  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749

This theorem is referenced by:  php  7608  zornn0g  8788  prn0  9272  genpn0  9286  nqpr  9297  ltexprlem5  9323  reclem2pr  9331  suplem1pr  9335  alexsubALTlem4  19757  bj-2upln0  32868  bj-2upln1upl  32869