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Theorem bnj882 29235
 Description: Definition (using hypotheses for readability) of the function giving the transitive closure of in by . (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj882.1
bnj882.2
bnj882.3
bnj882.4
Assertion
Ref Expression
bnj882
Distinct variable groups:   ,,,,   ,,,,   ,,,,
Allowed substitution hints:   (,,,)   (,,,)   (,,,)   (,,,)

Proof of Theorem bnj882
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-bnj18 28997 . 2
2 df-iun 4088 . . 3
3 df-iun 4088 . . . 4
4 bnj882.4 . . . . . . . . 9
5 bnj882.3 . . . . . . . . . . 11
6 bnj882.1 . . . . . . . . . . . . . 14
7 bnj882.2 . . . . . . . . . . . . . 14
86, 7anbi12i 679 . . . . . . . . . . . . 13
98anbi2i 676 . . . . . . . . . . . 12
10 3anass 940 . . . . . . . . . . . 12
11 3anass 940 . . . . . . . . . . . 12
129, 10, 113bitr4i 269 . . . . . . . . . . 11
135, 12rexeqbii 2729 . . . . . . . . . 10
1413abbii 2548 . . . . . . . . 9
154, 14eqtri 2456 . . . . . . . 8
1615eleq2i 2500 . . . . . . 7
1716anbi1i 677 . . . . . 6
1817rexbii2 2727 . . . . 5
1918abbii 2548 . . . 4
203, 19eqtr4i 2459 . . 3
212, 20eqtr4i 2459 . 2
221, 21eqtr4i 2459 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  cab 2422  wral 2698  wrex 2699   cdif 3310  c0 3621  csn 3807  ciun 4086   csuc 4576  com 4838   cdm 4871   wfn 5442  cfv 5447   c-bnj14 28990   c-bnj18 28996 This theorem is referenced by:  bnj893  29237  bnj906  29239  bnj916  29242  bnj983  29260  bnj1014  29269  bnj1145  29300  bnj1318  29332 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-rex 2704  df-iun 4088  df-bnj18 28997
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