ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  unssin Structured version   GIF version

Theorem unssin 3170
Description: Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
Assertion
Ref Expression
unssin (AB) ⊆ (V ∖ ((V ∖ A) ∩ (V ∖ B)))

Proof of Theorem unssin
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 oranim 806 . . . . 5 ((x A x B) → ¬ (¬ x A ¬ x B))
2 eldifn 3061 . . . . . 6 (x (V ∖ A) → ¬ x A)
3 eldifn 3061 . . . . . 6 (x (V ∖ B) → ¬ x B)
42, 3anim12i 321 . . . . 5 ((x (V ∖ A) x (V ∖ B)) → (¬ x A ¬ x B))
51, 4nsyl 558 . . . 4 ((x A x B) → ¬ (x (V ∖ A) x (V ∖ B)))
6 elin 3120 . . . 4 (x ((V ∖ A) ∩ (V ∖ B)) ↔ (x (V ∖ A) x (V ∖ B)))
75, 6sylnibr 601 . . 3 ((x A x B) → ¬ x ((V ∖ A) ∩ (V ∖ B)))
8 elun 3078 . . 3 (x (AB) ↔ (x A x B))
9 vex 2554 . . . 4 x V
10 eldif 2921 . . . 4 (x (V ∖ ((V ∖ A) ∩ (V ∖ B))) ↔ (x V ¬ x ((V ∖ A) ∩ (V ∖ B))))
119, 10mpbiran 846 . . 3 (x (V ∖ ((V ∖ A) ∩ (V ∖ B))) ↔ ¬ x ((V ∖ A) ∩ (V ∖ B)))
127, 8, 113imtr4i 190 . 2 (x (AB) → x (V ∖ ((V ∖ A) ∩ (V ∖ B))))
1312ssriv 2943 1 (AB) ⊆ (V ∖ ((V ∖ A) ∩ (V ∖ B)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 628   wcel 1390  Vcvv 2551  cdif 2908  cun 2909  cin 2910  wss 2911
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator