Theorem npss0 2305

Description: No set is a proper subset of the empty set.

Assertion

Ref	Expression
npss0	|- -. A (. (/)

Proof of Theorem npss0

Step	Hyp	Ref	Expression
1		eqid 1473	. 2 |- (/) = (/)
2		pssss 2139	. . . . 5 |- (A (. (/) -> A (_ (/))
3		ss0 2299	. . . . 5 |- (A (_ (/) -> A = (/))
4		psseq1 2131	. . . . 5 |- (A = (/) -> (A (. (/) <-> (/) (. (/)))
5	2, 3, 4	3syl 20	. . . 4 |- (A (. (/) -> (A (. (/) <-> (/) (. (/)))
6	5	ibi 591	. . 3 |- (A (. (/) -> (/) (. (/))
7		0pss 2304	. . . 4 |- ((/) (. (/) <-> (/) =/= (/))
8		df-ne 1584	. . . 4 |- ((/) =/= (/) <-> -. (/) = (/))
9	7, 8	bitr 173	. . 3 |- ((/) (. (/) <-> -. (/) = (/))
10	6, 9	sylib 198	. 2 |- (A (. (/) -> -. (/) = (/))
11	1, 10	mt2 109	1 |- -. A (. (/)

Syntax hints:  -. wn 2   <-> wb 146   = wceq 954   =/= wne 1582   (_ wss 2043   (. wpss 2044  (/)c0 2276

This theorem is referenced by:  pssnn 4519

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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