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Theorem nss 3329
 Description: Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
Assertion
Ref Expression
nss A Bx(x A ¬ x B))
Distinct variable groups:   x,A   x,B

Proof of Theorem nss
StepHypRef Expression
1 exanali 1585 . . 3 (x(x A ¬ x B) ↔ ¬ x(x Ax B))
2 dfss2 3262 . . 3 (A Bx(x Ax B))
31, 2xchbinxr 302 . 2 (x(x A ¬ x B) ↔ ¬ A B)
43bicomi 193 1 A Bx(x A ¬ x B))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710   ⊆ wss 3257 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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