ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nssinpss Structured version   GIF version

Theorem nssinpss 3146
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 3134 . . 3 (AB) ⊆ A
21biantrur 287 . 2 ((AB) ≠ A ↔ ((AB) ⊆ A (AB) ≠ A))
3 df-ss 2908 . . 3 (AB ↔ (AB) = A)
43necon3bbii 2220 . 2 AB ↔ (AB) ≠ A)
5 df-pss 2910 . 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 2186  cin 2893  wss 2894  wpss 2895
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-in 2901  df-ss 2908  df-pss 2910
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator