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Theorem setlikespec 25447
 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 2951 . . . . . 6
21elpred 25437 . . . . 5
32adantr 452 . . . 4 Se
43abbi2dv 2550 . . 3 Se
5 df-rab 2706 . . 3
64, 5syl6reqr 2486 . 2 Se
7 seex 4537 . . 3 Se
87ancoms 440 . 2 Se
96, 8eqeltrrd 2510 1 Se
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wcel 1725  cab 2421  crab 2701  cvv 2948   class class class wbr 4204   Se wse 4531  cpred 25426 This theorem is referenced by:  trpredtr  25493  trpredmintr  25494  trpredelss  25495  dftrpred3g  25496  trpredpo  25498  trpredrec  25501  frmin  25502  wfrlem15  25537 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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-se 4534  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-pred 25427
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