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Theorem predep 25472
 Description: The predecessor under the epsilon relationship is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
predep

Proof of Theorem predep
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-pred 25444 . 2
2 relcnv 5245 . . . . 5
3 relimasn 5230 . . . . 5
42, 3ax-mp 5 . . . 4
5 vex 2961 . . . . . . 7
6 brcnvg 5056 . . . . . . 7
75, 6mpan2 654 . . . . . 6
8 epelg 4498 . . . . . 6
97, 8bitrd 246 . . . . 5
109abbi1dv 2554 . . . 4
114, 10syl5eq 2482 . . 3
1211ineq2d 3544 . 2
131, 12syl5eq 2482 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wceq 1653   wcel 1726  cab 2424  cvv 2958   cin 3321  csn 3816   class class class wbr 4215   cep 4495  ccnv 4880  cima 4884   wrel 4886  cpred 25443 This theorem is referenced by:  predon  25473  epsetlike  25474  omsinds  25499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pr 4406 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 4216  df-opab 4270  df-eprel 4497  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-pred 25444
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