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Theorem nssssr 4082
Description: Negation of subclass relationship. Compare nssr 3107. (Contributed by Jim Kingdon, 17-Sep-2018.)
Assertion
Ref Expression
nssssr (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ 𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nssssr
StepHypRef Expression
1 exanaliim 1594 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
2 ssextss 4080 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2sylnibr 643 1 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wal 1297  wex 1436  wss 3021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480
This theorem is referenced by: (None)
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