Theorem 0pss 2308

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 2055	. . 3 |- ((/) (. A <-> ((/) (_ A /\ (/) =/= A))
2		0ss 2301	. . 3 |- (/) (_ A
3	1, 2	mpbiran 728	. 2 |- ((/) (. A <-> (/) =/= A)
4		necom 1636	. 2 |- ((/) =/= A <-> A =/= (/))
5	3, 4	bitr 173	1 |- ((/) (. A <-> A =/= (/))

Syntax hints:   <-> wb 146   =/= wne 1585   (_ wss 2047   (. wpss 2048  (/)c0 2280

This theorem is referenced by:  npss0 2309  php 4513  prn0 5093  genpn0 5106  1pr 5117  ltexprlem5 5146  reclem1pr 5156  suplem1pr 5161  infxpidmlem10 7561

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281

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