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Theorem 0pss 4239
 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 4198 . . 3 ∅ ⊆ 𝐴
2 df-pss 3808 . . 3 (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
31, 2mpbiran 699 . 2 (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4 necom 3022 . 2 (∅ ≠ 𝐴𝐴 ≠ ∅)
53, 4bitri 267 1 (∅ ⊊ 𝐴𝐴 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ≠ wne 2969   ⊆ wss 3792   ⊊ wpss 3793  ∅c0 4141 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-ne 2970  df-dif 3795  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142 This theorem is referenced by:  php  8432  zornn0g  9662  prn0  10146  genpn0  10160  nqpr  10171  ltexprlem5  10197  reclem2pr  10205  suplem1pr  10209  alexsubALTlem4  22262  bj-2upln0  33583  bj-2upln1upl  33584  0pssin  39020
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