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Theorem bnj882 29809
 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 29572 . 2
2 df-iun 4271 . . 3
3 df-iun 4271 . . . 4
4 bnj882.4 . . . . . . . . 9
5 bnj882.3 . . . . . . . . . . 11
6 bnj882.1 . . . . . . . . . . . . . 14
7 bnj882.2 . . . . . . . . . . . . . 14
86, 7anbi12i 711 . . . . . . . . . . . . 13
98anbi2i 708 . . . . . . . . . . . 12
10 3anass 1011 . . . . . . . . . . . 12
11 3anass 1011 . . . . . . . . . . . 12
129, 10, 113bitr4i 285 . . . . . . . . . . 11
135, 12rexeqbii 2894 . . . . . . . . . 10
1413abbii 2587 . . . . . . . . 9
154, 14eqtri 2493 . . . . . . . 8
1615eleq2i 2541 . . . . . . 7
1716anbi1i 709 . . . . . 6
1817rexbii2 2879 . . . . 5
1918abbii 2587 . . . 4
203, 19eqtr4i 2496 . . 3
212, 20eqtr4i 2496 . 2
221, 21eqtr4i 2496 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   w3a 1007   wceq 1452   wcel 1904  cab 2457  wral 2756  wrex 2757   cdif 3387  c0 3722  csn 3959  ciun 4269   cdm 4839   csuc 5432   wfn 5584  cfv 5589  com 6711   c-bnj14 29565   c-bnj18 29571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-rex 2762  df-iun 4271  df-bnj18 29572 This theorem is referenced by:  bnj893  29811  bnj906  29813  bnj916  29816  bnj983  29834  bnj1014  29843  bnj1145  29874  bnj1318  29906
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