Theorem npss0OLD 2912

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

Assertion

Ref	Expression
npss0OLD	|- -. A C. (/)

Proof of Theorem npss0OLD

Step	Hyp	Ref	Expression
1		eqid 1884	. 2 |- (/) = (/)
2		pssss 2705	. . . . 5 |- (A C. (/) -> A C_ (/))
3		ss0 2902	. . . . 5 |- (A C_ (/) -> A = (/))
4		psseq1 2697	. . . . 5 |- (A = (/) -> (A C. (/) <-> (/) C. (/)))
5	2, 3, 4	3syl 24	. . . 4 |- (A C. (/) -> (A C. (/) <-> (/) C. (/)))
6	5	ibi 652	. . 3 |- (A C. (/) -> (/) C. (/))
7		0pss 2910	. . . 4 |- ((/) C. (/) <-> (/) =/= (/))
8		df-ne 2019	. . . 4 |- ((/) =/= (/) <-> -. (/) = (/))
9	7, 8	bitri 190	. . 3 |- ((/) C. (/) <-> -. (/) = (/))
10	6, 9	sylib 215	. 2 |- (A C. (/) -> -. (/) = (/))
11	1, 10	mt2 124	1 |- -. A C. (/)

Syntax hints:  -. wn 2   <-> wb 163   = wceq 1298   =/= wne 2017   C_ wss 2593   C. wpss 2594  (/)c0 2875

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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