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Theorem ssconb 3573
 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 3434 . . . . . . 7
2 ssel 3434 . . . . . . 7
3 pm5.1 853 . . . . . . 7
41, 2, 3syl2an 477 . . . . . 6
5 con2b 334 . . . . . . 7
65a1i 11 . . . . . 6
74, 6anbi12d 710 . . . . 5
8 jcab 858 . . . . 5
9 jcab 858 . . . . 5
107, 8, 93bitr4g 288 . . . 4
11 eldif 3422 . . . . 5
1211imbi2i 312 . . . 4
13 eldif 3422 . . . . 5
1413imbi2i 312 . . . 4
1510, 12, 143bitr4g 288 . . 3
1615albidv 1680 . 2
17 dfss2 3429 . 2
18 dfss2 3429 . 2
1916, 17, 183bitr4g 288 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wa 369  wal 1368   wcel 1757   cdif 3409   wss 3412 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-v 3056  df-dif 3415  df-in 3419  df-ss 3426 This theorem is referenced by:  pssdifcom1  3848  pssdifcom2  3849  sbthlem1  7507  sbthlem2  7508  rpnnen2lem11  13595  setscom  14292  dpjidcl  16648  dpjidclOLD  16655  clsval2  18756  regsep2  19082  ordtconlem1  26474  conss2  29823
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