Theorem npss0 1731

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		cleqid 1102	. 2 |- (/) = (/)
2		pssss 1567	. . . . 5 |- (A (. (/) -> A (_ (/))
3		ss0 1727	. . . . 5 |- (A (_ (/) -> A = (/))
4		psseq1 1559	. . . . 5 |- (A = (/) -> (A (. (/) <-> (/) (. (/)))
5	2, 3, 4	3syl 21	. . . 4 |- (A (. (/) -> (A (. (/) <-> (/) (. (/)))
6	5	ibi 449	. . 3 |- (A (. (/) -> (/) (. (/))
7		0pss 1730	. . 3 |- ((/) (. (/) <-> -. (/) = (/))
8	6, 7	sylib 173	. 2 |- (A (. (/) -> -. (/) = (/))
9	1, 8	mt2 96	1 |- -. A (. (/)

Syntax hints:  -. wn 1   <-> wb 127   = wceq 1091   (_ wss 1487   (. wpss 1488  (/)c0 1707

This theorem is referenced by:  pssnn 3428

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708

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