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Theorem nssr 2980
 Description: Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)
Assertion
Ref Expression
nssr (x(x A ¬ x B) → ¬ AB)
Distinct variable groups:   x,A   x,B

Proof of Theorem nssr
StepHypRef Expression
1 exanaliim 1520 . 2 (x(x A ¬ x B) → ¬ x(x Ax B))
2 dfss2 2911 . 2 (ABx(x Ax B))
31, 2sylnibr 589 1 (x(x A ¬ x B) → ¬ AB)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wal 1226  ∃wex 1362   ∈ wcel 1374   ⊆ wss 2894 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-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  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-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908 This theorem is referenced by: (None)
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