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Theorem sscon 3973
Description: Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
sscon (𝐴𝐵 → (𝐶𝐵) ⊆ (𝐶𝐴))

Proof of Theorem sscon
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel 3821 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21con3d 150 . . . 4 (𝐴𝐵 → (¬ 𝑥𝐵 → ¬ 𝑥𝐴))
32anim2d 605 . . 3 (𝐴𝐵 → ((𝑥𝐶 ∧ ¬ 𝑥𝐵) → (𝑥𝐶 ∧ ¬ 𝑥𝐴)))
4 eldif 3808 . . 3 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐵))
5 eldif 3808 . . 3 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
63, 4, 53imtr4g 288 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐶𝐵) → 𝑥 ∈ (𝐶𝐴)))
76ssrdv 3833 1 (𝐴𝐵 → (𝐶𝐵) ⊆ (𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wcel 2164  cdif 3795  wss 3798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-dif 3801  df-in 3805  df-ss 3812
This theorem is referenced by:  sscond  3976  complss  3980  sorpsscmpl  7213  sbthlem1  8345  sbthlem2  8346  cantnfp1lem1  8859  cantnfp1lem3  8861  isf34lem7  9523  isf34lem6  9524  setsres  16271  mplsubglem  19802  cctop  21188  clsval2  21232  ntrss  21237  hauscmplem  21587  ptbasin  21758  cfinfil  22074  csdfil  22075  uniioombllem5  23760  kur14lem6  31735  bj-2upln1upl  33529  dvasin  34034  sscon34b  39152  clsk3nimkb  39173  fourierdlem62  41173  caragendifcl  41516
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