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Theorem uniuni 6609
Description: Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)
Assertion
Ref Expression
uniuni
Distinct variable group:   , ,

Proof of Theorem uniuni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 4252 . . . . . 6
21anbi2i 694 . . . . 5
32exbii 1667 . . . 4
4 19.42v 1775 . . . . . . 7
54bicomi 202 . . . . . 6
65exbii 1667 . . . . 5
7 excom 1849 . . . . . 6
8 anass 649 . . . . . . . 8
9 ancom 450 . . . . . . . 8
108, 9bitr3i 251 . . . . . . 7
11102exbii 1668 . . . . . 6
12 exdistr 1776 . . . . . 6
137, 11, 123bitri 271 . . . . 5
14 eluni 4252 . . . . . . . 8
1514bicomi 202 . . . . . . 7
1615anbi2i 694 . . . . . 6
1716exbii 1667 . . . . 5
186, 13, 173bitri 271 . . . 4
19 vex 3112 . . . . . . . . . . 11
2019uniex 6596 . . . . . . . . . 10
21 eleq2 2530 . . . . . . . . . 10
2220, 21ceqsexv 3146 . . . . . . . . 9
23 exancom 1671 . . . . . . . . 9
2422, 23bitr3i 251 . . . . . . . 8
2524anbi2i 694 . . . . . . 7
26 19.42v 1775 . . . . . . 7
27 ancom 450 . . . . . . . . 9
28 anass 649 . . . . . . . . 9
2927, 28bitri 249 . . . . . . . 8
3029exbii 1667 . . . . . . 7
3125, 26, 303bitr2i 273 . . . . . 6
3231exbii 1667 . . . . 5
33 excom 1849 . . . . 5
34 exdistr 1776 . . . . . 6
35 vex 3112 . . . . . . . . . 10
36 eqeq1 2461 . . . . . . . . . . . 12
3736anbi1d 704 . . . . . . . . . . 11
3837exbidv 1714 . . . . . . . . . 10
3935, 38elab 3246 . . . . . . . . 9
4039bicomi 202 . . . . . . . 8
4140anbi2i 694 . . . . . . 7
4241exbii 1667 . . . . . 6
4334, 42bitri 249 . . . . 5
4432, 33, 433bitri 271 . . . 4
453, 18, 443bitri 271 . . 3
4645abbii 2591 . 2
47 df-uni 4250 . 2
48 df-uni 4250 . 2
4946, 47, 483eqtr4i 2496 1
Colors of variables: wff setvar class
Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  U.cuni 4249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-v 3111  df-uni 4250
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