Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  npss0 Structured version   GIF version

Theorem npss0 3243
 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 3232 . . . 4 ∅ ⊆ A
21a1i 9 . . 3 (A ⊆ ∅ → ∅ ⊆ A)
3 imanim 778 . . 3 ((A ⊆ ∅ → ∅ ⊆ A) → ¬ (A ⊆ ∅ ¬ ∅ ⊆ A))
42, 3ax-mp 7 . 2 ¬ (A ⊆ ∅ ¬ ∅ ⊆ A)
5 dfpss3 3007 . 2 (A ⊊ ∅ ↔ (A ⊆ ∅ ¬ ∅ ⊆ A))
64, 5mtbir 583 1 ¬ A ⊊ ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ⊆ wss 2894   ⊊ wpss 2895  ∅c0 3201 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908  df-pss 2910  df-nul 3202 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator