Theorem 0pss 4013

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 3972	. . 3 |- (/) C_ A
2		df-pss 3590	. . 3 |- ( (/) C. A <-> ( (/) C_ A /\ (/) =/= A ) )
3	1, 2	mpbiran 953	. 2 |- ( (/) C. A <-> (/) =/= A )
4		necom 2847	. 2 |- ( (/) =/= A <-> A =/= (/) )
5	3, 4	bitri 264	1 |- ( (/) C. A <-> A =/= (/) )

Syntax hints:   <-> wb 196   =/= wne 2794   C_ wss 3574   C. wpss 3575   (/)c0 3915

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-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-in 3581  df-ss 3588  df-pss 3590  df-nul 3916

This theorem is referenced by:  php  8144  zornn0g  9327  prn0  9811  genpn0  9825  nqpr  9836  ltexprlem5  9862  reclem2pr  9870  suplem1pr  9874  alexsubALTlem4  21854  bj-2upln0  33011  bj-2upln1upl  33012  0pssin  38064