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Theorem npss0 3589
 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

Proof of Theorem npss0
StepHypRef Expression
1 0ss 3579 . . . 4 A
21a1i 10 . . 3 (A A)
3 iman 413 . . 3 ((A A) ↔ ¬ (A ¬ A))
42, 3mpbi 199 . 2 ¬ (A ¬ A)
5 dfpss3 3355 . 2 (A ↔ (A ¬ A))
64, 5mtbir 290 1 ¬ A
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ⊆ wss 3257   ⊊ wpss 3258  ∅c0 3550 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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-pss 3261  df-nul 3551 This theorem is referenced by: (None)
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