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Theorem ssconb 3472
 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 3334 . . . . . . 7
2 ssel 3334 . . . . . . 7
3 pm5.1 831 . . . . . . 7
41, 2, 3syl2an 464 . . . . . 6
5 con2b 325 . . . . . . 7
65a1i 11 . . . . . 6
74, 6anbi12d 692 . . . . 5
8 jcab 834 . . . . 5
9 jcab 834 . . . . 5
107, 8, 93bitr4g 280 . . . 4
11 eldif 3322 . . . . 5
1211imbi2i 304 . . . 4
13 eldif 3322 . . . . 5
1413imbi2i 304 . . . 4
1510, 12, 143bitr4g 280 . . 3
1615albidv 1635 . 2
17 dfss2 3329 . 2
18 dfss2 3329 . 2
1916, 17, 183bitr4g 280 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359  wal 1549   wcel 1725   cdif 3309   wss 3312 This theorem is referenced by:  pssdifcom1  3705  pssdifcom2  3706  sbthlem1  7209  sbthlem2  7210  rpnnen2lem11  12816  setscom  13489  dpjidcl  15608  clsval2  17106  regsep2  17432  conss2  27603 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326
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