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Theorem ssconb 3399
 Description: Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
ssconb

Proof of Theorem ssconb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . . . . . 7
2 ssel 3267 . . . . . . 7
3 pm5.1 830 . . . . . . 7
41, 2, 3syl2an 463 . . . . . 6
5 con2b 324 . . . . . . 7
65a1i 10 . . . . . 6
74, 6anbi12d 691 . . . . 5
8 jcab 833 . . . . 5
9 jcab 833 . . . . 5
107, 8, 93bitr4g 279 . . . 4
11 eldif 3221 . . . . 5
1211imbi2i 303 . . . 4
13 eldif 3221 . . . . 5
1413imbi2i 303 . . . 4
1510, 12, 143bitr4g 279 . . 3
1615albidv 1625 . 2
17 dfss2 3262 . 2
18 dfss2 3262 . 2
1916, 17, 183bitr4g 279 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358  wal 1540   wcel 1710   cdif 3206   wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-dif 3215  df-ss 3259 This theorem is referenced by:  pssdifcom1  3635  pssdifcom2  3636
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