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Theorem setlikespec 29235
 Description: If is set-like in , then all predecessors classes of elements of exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
setlikespec Se

Proof of Theorem setlikespec
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3096 . . . . . 6
21elpred 29225 . . . . 5
32adantr 465 . . . 4 Se
43abbi2dv 2578 . . 3 Se
5 df-rab 2800 . . 3
64, 5syl6reqr 2501 . 2 Se
7 seex 4828 . . 3 Se
87ancoms 453 . 2 Se
96, 8eqeltrrd 2530 1 Se
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wcel 1802  cab 2426  crab 2795  cvv 3093   class class class wbr 4433   Se wse 4822  cpred 29211 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-se 4825  df-xp 4991  df-cnv 4993  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-pred 29212 This theorem is referenced by:  trpredtr  29281  trpredmintr  29282  trpredelss  29283  dftrpred3g  29284  trpredpo  29286  trpredrec  29289  frmin  29290  wfrlem15  29325
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