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Theorem nssss 4954
Description: Negation of subclass relationship. Compare nss 3696. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
nssss 𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nssss
StepHypRef Expression
1 exanali 1826 . . 3 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
2 ssextss 4952 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2xchbinxr 324 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ 𝐴𝐵)
43bicomi 214 1 𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521  wex 1744  wss 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213
This theorem is referenced by: (None)
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