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Theorem nssinpss 4186
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 4158 . . 3 (𝐴𝐵) ⊆ 𝐴
21biantrur 534 . 2 ((𝐴𝐵) ≠ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
3 df-ss 3901 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
43necon3bbii 3037 . 2 𝐴𝐵 ↔ (𝐴𝐵) ≠ 𝐴)
5 df-pss 3903 . 2 ((𝐴𝐵) ⊊ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
62, 4, 53bitr4i 306 1 𝐴𝐵 ↔ (𝐴𝐵) ⊊ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wne 2990  cin 3883  wss 3884  wpss 3885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-v 3446  df-in 3891  df-ss 3901  df-pss 3903
This theorem is referenced by:  fbfinnfr  22449  chrelat2i  30151
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