ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmuni Structured version   Unicode version

Theorem dmuni 4460
Description: The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
Assertion
Ref Expression
dmuni  dom  U.  U_  dom
Distinct variable group:   ,

Proof of Theorem dmuni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 excom 1527 . . . . 5  <. ,  >.  <. , 
>.
2 ancom 253 . . . . . . 7  <. , 
>.  <. , 
>.
3 19.41v 1755 . . . . . . 7  <. , 
>.  <. , 
>.
4 vex 2529 . . . . . . . . 9 
_V
54eldm2 4448 . . . . . . . 8  dom  <. ,  >.
65anbi2i 430 . . . . . . 7  dom  <. ,  >.
72, 3, 63bitr4i 201 . . . . . 6  <. , 
>.  dom
87exbii 1469 . . . . 5  <. ,  >.  dom
91, 8bitri 173 . . . 4  <. ,  >.  dom
10 eluni 3546 . . . . 5  <. ,  >.  U.  <. ,  >.
1110exbii 1469 . . . 4  <. , 
>.  U.  <. ,  >.
12 df-rex 2281 . . . 4  dom  dom
139, 11, 123bitr4i 201 . . 3  <. , 
>.  U.  dom
144eldm2 4448 . . 3  dom  U.  <. , 
>.  U.
15 eliun 3624 . . 3  U_  dom  dom
1613, 14, 153bitr4i 201 . 2  dom  U.  U_  dom
1716eqriv 2010 1  dom  U.  U_  dom
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1223  wex 1354   wcel 1366  wrex 2276   <.cop 3342   U.cuni 3543   U_ciun 3620   dom cdm 4260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-iun 3622  df-br 3728  df-dm 4270
This theorem is referenced by:  tfrlem8  5844  tfrlemi14d  5856  tfrlemi14  5857
  Copyright terms: Public domain W3C validator