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Theorem preddowncl 25473
 Description: A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.)
Assertion
Ref Expression
preddowncl
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem preddowncl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2498 . . . . 5
2 predeq3 25445 . . . . . 6
3 predeq3 25445 . . . . . 6
42, 3eqeq12d 2452 . . . . 5
51, 4imbi12d 313 . . . 4
65imbi2d 309 . . 3
7 predpredss 25447 . . . . . 6
87ad2antrr 708 . . . . 5
9 predeq3 25445 . . . . . . . . . . . 12
109sseq1d 3377 . . . . . . . . . . 11
1110rspccva 3053 . . . . . . . . . 10
1211sseld 3349 . . . . . . . . 9
13 vex 2961 . . . . . . . . . . 11
1413elpredim 25453 . . . . . . . . . 10
1514a1i 11 . . . . . . . . 9
1612, 15jcad 521 . . . . . . . 8
17 vex 2961 . . . . . . . . . . 11
1817elpred 25454 . . . . . . . . . 10
1918imbi2d 309 . . . . . . . . 9
2019adantl 454 . . . . . . . 8
2116, 20mpbird 225 . . . . . . 7
2221ssrdv 3356 . . . . . 6
2322adantll 696 . . . . 5
248, 23eqssd 3367 . . . 4
2524ex 425 . . 3
266, 25vtoclg 3013 . 2
2726pm2.43b 49 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707   wss 3322   class class class wbr 4214  cpred 25440 This theorem is referenced by:  wfrlem4  25543  frrlem4  25587 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 25441
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