Theorem 0pss 2304

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

Assertion

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

Proof of Theorem 0pss

Step	Hyp	Ref	Expression
1		df-pss 2051	. . 3 |- ((/) (. A <-> ((/) (_ A /\ (/) =/= A))
2		0ss 2297	. . 3 |- (/) (_ A
3	1, 2	mpbiran 727	. 2 |- ((/) (. A <-> (/) =/= A)
4		necom 1633	. 2 |- ((/) =/= A <-> A =/= (/))
5	3, 4	bitr 173	1 |- ((/) (. A <-> A =/= (/))

Syntax hints:   <-> wb 146   =/= wne 1582   (_ wss 2043   (. wpss 2044  (/)c0 2276

This theorem is referenced by:  npss0 2305  php 4499  prn0 5073  genpn0 5086  1pr 5097  ltexprlem5 5126  reclem1pr 5136  suplem1pr 5141  infxpidmlem10 7512

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277

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