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Theorem nssinpss 3487
 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 A B ↔ (AB) ⊊ A)

Proof of Theorem nssinpss
StepHypRef Expression
1 inss1 3475 . . 3 (AB) A
21biantrur 492 . 2 ((AB) ≠ A ↔ ((AB) A (AB) ≠ A))
3 df-ss 3259 . . 3 (A B ↔ (AB) = A)
43necon3bbii 2547 . 2 A B ↔ (AB) ≠ A)
5 df-pss 3261 . 2 ((AB) ⊊ A ↔ ((AB) A (AB) ≠ A))
62, 4, 53bitr4i 268 1 A B ↔ (AB) ⊊ A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ≠ wne 2516   ∩ cin 3208   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261 This theorem is referenced by: (None)
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