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Theorem nssinpss 4159
 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 4131 . . 3 (𝐴𝐵) ⊆ 𝐴
21biantrur 531 . 2 ((𝐴𝐵) ≠ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
3 df-ss 3880 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
43necon3bbii 3033 . 2 𝐴𝐵 ↔ (𝐴𝐵) ≠ 𝐴)
5 df-pss 3882 . 2 ((𝐴𝐵) ⊊ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
62, 4, 53bitr4i 304 1 𝐴𝐵 ↔ (𝐴𝐵) ⊊ 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ≠ wne 2986   ∩ cin 3864   ⊆ wss 3865   ⊊ wpss 3866 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-v 3442  df-in 3872  df-ss 3880  df-pss 3882 This theorem is referenced by:  fbfinnfr  22137  chrelat2i  29829
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