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Theorem predep 25406
 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 25382 . 2
2 relcnv 5201 . . . . 5
3 relimasn 5186 . . . . 5
42, 3ax-mp 8 . . . 4
5 vex 2919 . . . . . . 7
6 brcnvg 5012 . . . . . . 7
75, 6mpan2 653 . . . . . 6
8 epelg 4455 . . . . . 6
97, 8bitrd 245 . . . . 5
109abbi1dv 2520 . . . 4
114, 10syl5eq 2448 . . 3
1211ineq2d 3502 . 2
131, 12syl5eq 2448 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1649   wcel 1721  cab 2390  cvv 2916   cin 3279  csn 3774   class class class wbr 4172   cep 4452  ccnv 4836  cima 4840   wrel 4842  cpred 25381 This theorem is referenced by:  predon  25407  epsetlike  25408  omsinds  25433 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-eprel 4454  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 25382
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