Theorem 0pss 3813

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 3774	. . 3 |- (/) C_ A
2		df-pss 3431	. . 3 |- ( (/) C. A <-> ( (/) C_ A /\ (/) =/= A ) )
3	1, 2	mpbiran 934	. 2 |- ( (/) C. A <-> (/) =/= A )
4		necom 2688	. 2 |- ( (/) =/= A <-> A =/= (/) )
5	3, 4	bitri 257	1 |- ( (/) C. A <-> A =/= (/) )

Syntax hints:   <-> wb 189   =/= wne 2632   C_ wss 3415   C. wpss 3416   (/)c0 3742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-v 3058  df-dif 3418  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743

This theorem is referenced by:  php  7781  zornn0g  8960  prn0  9439  genpn0  9453  nqpr  9464  ltexprlem5  9490  reclem2pr  9498  suplem1pr  9502  alexsubALTlem4  21113  bj-2upln0  31661  bj-2upln1upl  31662  0pssin  36409