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Theorem sscon34 3661
 Description: Contraposition law for subset. (Contributed by SF, 11-Mar-2015.)
Assertion
Ref Expression
sscon34 (A B ↔ ∼ B A)

Proof of Theorem sscon34
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 con34b 283 . . . 4 ((x Ax B) ↔ (¬ x B → ¬ x A))
2 vex 2862 . . . . . 6 x V
32elcompl 3225 . . . . 5 (x B ↔ ¬ x B)
42elcompl 3225 . . . . 5 (x A ↔ ¬ x A)
53, 4imbi12i 316 . . . 4 ((x Bx A) ↔ (¬ x B → ¬ x A))
61, 5bitr4i 243 . . 3 ((x Ax B) ↔ (x Bx A))
76albii 1566 . 2 (x(x Ax B) ↔ x(x Bx A))
8 dfss2 3262 . 2 (A Bx(x Ax B))
9 dfss2 3262 . 2 ( ∼ B Ax(x Bx A))
107, 8, 93bitr4i 268 1 (A B ↔ ∼ B A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710   ∼ ccompl 3205   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  sbthlem1  6203
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