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Theorem 0pss 3991
 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 3950 . . 3 ∅ ⊆ 𝐴
2 df-pss 3576 . . 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 3560   ⊊ wpss 3561  ∅c0 3897 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 3192  df-dif 3563  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898 This theorem is referenced by:  php  8104  zornn0g  9287  prn0  9771  genpn0  9785  nqpr  9796  ltexprlem5  9822  reclem2pr  9830  suplem1pr  9834  alexsubALTlem4  21794  bj-2upln0  32711  bj-2upln1upl  32712  0pssin  37585
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