Theorem 0pss 2910

Description: The null set is a proper subset of any non-empty set.

Assertion

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

Proof of Theorem 0pss

Step	Hyp	Ref	Expression
1		df-pss 2607	. . 3 |- ((/) C. A <-> ((/) C_ A /\ (/) =/= A))
2		0ss 2900	. . 3 |- (/) C_ A
3	1, 2	mpbiran 798	. 2 |- ((/) C. A <-> (/) =/= A)
4		necom 2094	. 2 |- ((/) =/= A <-> A =/= (/))
5	3, 4	bitri 190	1 |- ((/) C. A <-> A =/= (/))

Syntax hints:   <-> wb 163   =/= wne 2017   C_ wss 2593   C. wpss 2594  (/)c0 2875

This theorem is referenced by:  npss0OLD 2912  php 5607  prn0 6245  genpn0 6258  1pr 6269  ltexprlem5 6298  reclem1pr 6308  suplem1pr 6313  infxpidmlem10 8830  alexsublem4 15440  filssufil 15571  zornn0 15764

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876

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