Theorem 0pss 3588

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

Assertion

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

Proof of Theorem 0pss

Step	Hyp	Ref	Expression
1		0ss 3579	. . 3 |- (/) C_ A
2		df-pss 3261	. . 3 |- ((/) C. A <-> ((/) C_ A /\ (/) =/= A))
3	1, 2	mpbiran 884	. 2 |- ((/) C. A <-> (/) =/= A)
4		necom 2597	. 2 |- ((/) =/= A <-> A =/= (/))
5	3, 4	bitri 240	1 |- ((/) C. A <-> A =/= (/))

Syntax hints:   <-> wb 176   =/= wne 2516   C_ wss 3257   C. wpss 3258  (/)c0 3550

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-pss 3261  df-nul 3551

This theorem is referenced by: (None)