Theorem npss0 2911

Description: No set is a proper subset of the empty set. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem npss0

Step	Hyp	Ref	Expression
1		0ss 2900	. . . 4 |- (/) C_ A
2	1	a1i 8	. . 3 |- (A C_ (/) -> (/) C_ A)
3		iman 256	. . 3 |- ((A C_ (/) -> (/) C_ A) <-> -. (A C_ (/) /\ -. (/) C_ A))
4	2, 3	mpbi 206	. 2 |- -. (A C_ (/) /\ -. (/) C_ A)
5		dfpss3 2695	. 2 |- (A C. (/) <-> (A C_ (/) /\ -. (/) C_ A))
6	4, 5	mtbir 209	1 |- -. A C. (/)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   C_ wss 2593   C. wpss 2594  (/)c0 2875

This theorem is referenced by:  pssnn 5628

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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