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Theorem nssinpss 4267
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 4237 . . 3 (𝐴𝐵) ⊆ 𝐴
21biantrur 530 . 2 ((𝐴𝐵) ≠ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
3 dfss2 3969 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
43necon3bbii 2988 . 2 𝐴𝐵 ↔ (𝐴𝐵) ≠ 𝐴)
5 df-pss 3971 . 2 ((𝐴𝐵) ⊊ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
62, 4, 53bitr4i 303 1 𝐴𝐵 ↔ (𝐴𝐵) ⊊ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wne 2940  cin 3950  wss 3951  wpss 3952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-in 3958  df-ss 3968  df-pss 3971
This theorem is referenced by:  fbfinnfr  23849  chrelat2i  32384
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