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Theorem nssinpss 3889
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 3866 . . 3 (𝐴𝐵) ⊆ 𝐴
21biantrur 526 . 2 ((𝐴𝐵) ≠ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
3 df-ss 3621 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
43necon3bbii 2870 . 2 𝐴𝐵 ↔ (𝐴𝐵) ≠ 𝐴)
5 df-pss 3623 . 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 383  wne 2823  cin 3606  wss 3607  wpss 3608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-in 3614  df-ss 3621  df-pss 3623
This theorem is referenced by:  fbfinnfr  21692  chrelat2i  29352
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