MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0pss Structured version   Visualization version   GIF version

Theorem 0pss 3985
Description: The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.)
Assertion
Ref Expression
0pss (∅ ⊊ 𝐴𝐴 ≠ ∅)

Proof of Theorem 0pss
StepHypRef Expression
1 0ss 3944 . . 3 ∅ ⊆ 𝐴
2 df-pss 3571 . . 3 (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
31, 2mpbiran 952 . 2 (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4 necom 2843 . 2 (∅ ≠ 𝐴𝐴 ≠ ∅)
53, 4bitri 264 1 (∅ ⊊ 𝐴𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wne 2790  wss 3555  wpss 3556  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892
This theorem is referenced by:  php  8088  zornn0g  9271  prn0  9755  genpn0  9769  nqpr  9780  ltexprlem5  9806  reclem2pr  9814  suplem1pr  9818  alexsubALTlem4  21764  bj-2upln0  32658  bj-2upln1upl  32659  0pssin  37546
  Copyright terms: Public domain W3C validator