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Theorem sscon 3071
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 2933 . . . . 5 (AB → (x Ax B))
21con3d 560 . . . 4 (AB → (¬ x B → ¬ x A))
32anim2d 320 . . 3 (AB → ((x 𝐶 ¬ x B) → (x 𝐶 ¬ x A)))
4 eldif 2921 . . 3 (x (𝐶B) ↔ (x 𝐶 ¬ x B))
5 eldif 2921 . . 3 (x (𝐶A) ↔ (x 𝐶 ¬ x A))
63, 4, 53imtr4g 194 . 2 (AB → (x (𝐶B) → x (𝐶A)))
76ssrdv 2945 1 (AB → (𝐶B) ⊆ (𝐶A))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wcel 1390  cdif 2908  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-bndl 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-in 2918  df-ss 2925
This theorem is referenced by:  sscond  3074
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