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Theorem nssinpss 3163
 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 3151 . . 3 (AB) ⊆ A
21biantrur 287 . 2 ((AB) ≠ A ↔ ((AB) ⊆ A (AB) ≠ A))
3 df-ss 2925 . . 3 (AB ↔ (AB) = A)
43necon3bbii 2236 . 2 AB ↔ (AB) ≠ A)
5 df-pss 2927 . 2 ((AB) ⊊ A ↔ ((AB) ⊆ A (AB) ≠ A))
62, 4, 53bitr4i 201 1 AB ↔ (AB) ⊊ A)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ≠ wne 2201   ∩ cin 2910   ⊆ wss 2911   ⊊ wpss 2912 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-in 2918  df-ss 2925  df-pss 2927 This theorem is referenced by: (None)
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