Theorem npss0 3590

Description: No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem npss0

Step	Hyp	Ref	Expression
1		0ss 3580	. . . 4 |- (/) C_ A
2	1	a1i 10	. . 3 |- (A C_ (/) -> (/) C_ A)
3		iman 413	. . 3 |- ((A C_ (/) -> (/) C_ A) <-> -. (A C_ (/) /\ -. (/) C_ A))
4	2, 3	mpbi 199	. 2 |- -. (A C_ (/) /\ -. (/) C_ A)
5		dfpss3 3356	. 2 |- (A C. (/) <-> (A C_ (/) /\ -. (/) C_ A))
6	4, 5	mtbir 290	1 |- -. A C. (/)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   C_ wss 3258   C. wpss 3259  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-pss 3262  df-nul 3552

This theorem is referenced by: (None)