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Theorem 0pss 4440
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 4392 . . 3 ∅ ⊆ 𝐴
2 df-pss 3964 . . 3 (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
31, 2mpbiran 708 . 2 (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4 necom 2990 . 2 (∅ ≠ 𝐴𝐴 ≠ ∅)
53, 4bitri 275 1 (∅ ⊊ 𝐴𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wne 2936  wss 3945  wpss 3946  c0 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-v 3472  df-dif 3948  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319
This theorem is referenced by:  php  9228  phpOLD  9240  zornn0g  10522  prn0  11006  genpn0  11020  nqpr  11031  ltexprlem5  11057  reclem2pr  11065  suplem1pr  11069  alexsubALTlem4  23947  bj-2upln0  36496  bj-2upln1upl  36497  0pssin  43195
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