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Theorem nssinpss 4005
Description: Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
nssinpss 𝐴𝐵 ↔ (𝐴𝐵) ⊊ 𝐴)

Proof of Theorem nssinpss
StepHypRef Expression
1 inss1 3981 . . 3 (𝐴𝐵) ⊆ 𝐴
21biantrur 514 . 2 ((𝐴𝐵) ≠ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
3 df-ss 3737 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
43necon3bbii 2990 . 2 𝐴𝐵 ↔ (𝐴𝐵) ≠ 𝐴)
5 df-pss 3739 . 2 ((𝐴𝐵) ⊊ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
62, 4, 53bitr4i 292 1 𝐴𝐵 ↔ (𝐴𝐵) ⊊ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wne 2943  cin 3722  wss 3723  wpss 3724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-in 3730  df-ss 3737  df-pss 3739
This theorem is referenced by:  fbfinnfr  21865  chrelat2i  29564
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