Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiun2g Unicode version

Theorem dfiun2g 4362
 Description: Alternate definition of indexed union when is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
dfiun2g
Distinct variable groups:   ,   ,   ,

Proof of Theorem dfiun2g
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfra1 2838 . . . . . 6
2 rsp 2823 . . . . . . . 8
3 clel3g 3237 . . . . . . . 8
42, 3syl6 33 . . . . . . 7
54imp 429 . . . . . 6
61, 5rexbida 2963 . . . . 5
7 rexcom4 3129 . . . . 5
86, 7syl6bb 261 . . . 4
9 r19.41v 3009 . . . . . 6
109exbii 1667 . . . . 5
11 exancom 1671 . . . . 5
1210, 11bitri 249 . . . 4
138, 12syl6bb 261 . . 3
14 eliun 4335 . . 3
15 eluniab 4260 . . 3
1613, 14, 153bitr4g 288 . 2
1716eqrdv 2454 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  A.wral 2807  E.wrex 2808  U.cuni 4249  U_ciun 4330 This theorem is referenced by:  dfiun2  4364  dfiun3g  5260  iunexg  6776  uniqs  7390  ac6num  8880  iunopn  19407  pnrmopn  19844  cncmp  19892  ptcmplem3  20554  iunmbl  21963  voliun  21964  sigaclcuni  28118  sigaclcu2  28120  sigaclci  28132  measvunilem  28183  meascnbl  28190 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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  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-ral 2812  df-rex 2813  df-v 3111  df-uni 4250  df-iun 4332
 Copyright terms: Public domain W3C validator