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Theorem 0pss 4144
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 4103 . . 3 ∅ ⊆ 𝐴
2 df-pss 3719 . . 3 (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
31, 2mpbiran 991 . 2 (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4 necom 2973 . 2 (∅ ≠ 𝐴𝐴 ≠ ∅)
53, 4bitri 264 1 (∅ ⊊ 𝐴𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wne 2920  wss 3703  wpss 3704  c0 4046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-v 3330  df-dif 3706  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047
This theorem is referenced by:  php  8297  zornn0g  9490  prn0  9974  genpn0  9988  nqpr  9999  ltexprlem5  10025  reclem2pr  10033  suplem1pr  10037  alexsubALTlem4  22026  bj-2upln0  33288  bj-2upln1upl  33289  0pssin  38535
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