Theorem npss0 2313

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 1478	. 2 |- (/) = (/)
2		pssss 2146	. . . . 5 |- (A (. (/) -> A (_ (/))
3		ss0 2307	. . . . 5 |- (A (_ (/) -> A = (/))
4		psseq1 2138	. . . . 5 |- (A = (/) -> (A (. (/) <-> (/) (. (/)))
5	2, 3, 4	3syl 20	. . . 4 |- (A (. (/) -> (A (. (/) <-> (/) (. (/)))
6	5	ibi 594	. . 3 |- (A (. (/) -> (/) (. (/))
7		0pss 2312	. . . 4 |- ((/) (. (/) <-> (/) =/= (/))
8		df-ne 1590	. . . 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 958   =/= wne 1588   (_ wss 2050   (. wpss 2051  (/)c0 2283

This theorem is referenced by:  pssnn 4544

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284

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