ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  0pss Structured version   GIF version

Theorem 0pss 3242
Description: The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)
Assertion
Ref Expression
0pss (∅ ⊊ AA ≠ ∅)

Proof of Theorem 0pss
StepHypRef Expression
1 0ss 3232 . . 3 ∅ ⊆ A
2 df-pss 2910 . . 3 (∅ ⊊ A ↔ (∅ ⊆ A ∅ ≠ A))
31, 2mpbiran 835 . 2 (∅ ⊊ A ↔ ∅ ≠ A)
4 necom 2267 . 2 (∅ ≠ AA ≠ ∅)
53, 4bitri 173 1 (∅ ⊊ AA ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wb 98  wne 2186  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