Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfss6 Structured version   Visualization version   GIF version

Theorem dfss6 37082
 Description: Another definition of subclasshood. (Contributed by RP, 16-Apr-2020.)
Assertion
Ref Expression
dfss6 (𝐴𝐵 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfss6
StepHypRef Expression
1 dfss2 3557 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2 notnotb 303 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ¬ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2bitri 263 . 2 (𝐴𝐵 ↔ ¬ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
4 exanali 1773 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
53, 4xchbinxr 324 1 (𝐴𝐵 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator