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Theorem sscon 4152
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 3988 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21con3d 152 . . . 4 (𝐴𝐵 → (¬ 𝑥𝐵 → ¬ 𝑥𝐴))
32anim2d 612 . . 3 (𝐴𝐵 → ((𝑥𝐶 ∧ ¬ 𝑥𝐵) → (𝑥𝐶 ∧ ¬ 𝑥𝐴)))
4 eldif 3972 . . 3 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐵))
5 eldif 3972 . . 3 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
63, 4, 53imtr4g 296 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐶𝐵) → 𝑥 ∈ (𝐶𝐴)))
76ssrdv 4000 1 (𝐴𝐵 → (𝐶𝐵) ⊆ (𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2105  cdif 3959  wss 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965  df-ss 3979
This theorem is referenced by:  sscond  4155  complss  4160  sscon34b  4309  sorpsscmpl  7752  sbthlem1  9121  sbthlem2  9122  cantnfp1lem1  9715  cantnfp1lem3  9717  isf34lem7  10416  isf34lem6  10417  setsres  17211  mplsubglem  22036  cctop  23028  clsval2  23073  ntrss  23078  hauscmplem  23429  ptbasin  23600  cfinfil  23916  csdfil  23917  uniioombllem5  25635  kur14lem6  35195  bj-2upln1upl  37006  dvasin  37690  readvrec2  42369  readvrec  42370  clsk3nimkb  44029  fourierdlem62  46123  caragendifcl  46469
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