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Theorem sorpsscmpl 6533
 Description: The componentwise complement of a chain of sets is also a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
sorpsscmpl [] []
Distinct variable groups:   ,   ,

Proof of Theorem sorpsscmpl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difeq2 3459 . . . . . . 7
21eleq1d 2502 . . . . . 6
32elrab 3092 . . . . 5
4 difeq2 3459 . . . . . . 7
54eleq1d 2502 . . . . . 6
65elrab 3092 . . . . 5
7 an4 798 . . . . . 6
87biimpi 187 . . . . 5
93, 6, 8syl2anb 466 . . . 4
10 sorpssi 6528 . . . . . . . 8 []
1110expcom 425 . . . . . . 7 []
12 vex 2959 . . . . . . . . . . . 12
1312elpw 3805 . . . . . . . . . . 11
14 dfss4 3575 . . . . . . . . . . 11
1513, 14bitri 241 . . . . . . . . . 10
16 vex 2959 . . . . . . . . . . . 12
1716elpw 3805 . . . . . . . . . . 11
18 dfss4 3575 . . . . . . . . . . 11
1917, 18bitri 241 . . . . . . . . . 10
20 sscon 3481 . . . . . . . . . . . 12
21 sseq12 3371 . . . . . . . . . . . 12
2220, 21syl5ib 211 . . . . . . . . . . 11
23 sscon 3481 . . . . . . . . . . . 12
24 sseq12 3371 . . . . . . . . . . . . 13
2524ancoms 440 . . . . . . . . . . . 12
2623, 25syl5ib 211 . . . . . . . . . . 11
2722, 26orim12d 812 . . . . . . . . . 10
2815, 19, 27syl2anb 466 . . . . . . . . 9
2928com12 29 . . . . . . . 8
3029orcoms 379 . . . . . . 7
3111, 30syl6 31 . . . . . 6 []
3231com3l 77 . . . . 5 []
3332imp3a 421 . . . 4 []
349, 33syl5 30 . . 3 []
3534ralrimivv 2797 . 2 []
36 sorpss 6527 . 2 []
3735, 36sylibr 204 1 [] []
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wo 358   wa 359   wceq 1652   wcel 1725  wral 2705  crab 2709   cdif 3317   wss 3320  cpw 3799   wor 4502   [] crpss 6521 This theorem is referenced by:  fin2i2  8198  isfin2-2  8199 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-rpss 6522
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