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Theorem nssr 3188
 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(𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ 𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nssr
StepHypRef Expression
1 exanaliim 1627 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
2 dfss2 3117 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2sylnibr 667 1 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ 𝐴𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1333  ∃wex 1472   ∈ wcel 2128   ⊆ wss 3102 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115 This theorem is referenced by: (None)
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