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

Theorem iunpwss 3872
 Description: Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
Assertion
Ref Expression
iunpwss
Distinct variable group:   ,

Proof of Theorem iunpwss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssiun 3823 . . 3
2 eliun 3785 . . . 4
3 vex 2661 . . . . . 6
43elpw 3484 . . . . 5
54rexbii 2417 . . . 4
62, 5bitri 183 . . 3
73elpw 3484 . . . 4
8 uniiun 3834 . . . . 5
98sseq2i 3092 . . . 4
107, 9bitri 183 . . 3
111, 6, 103imtr4i 200 . 2
1211ssriv 3069 1
 Colors of variables: wff set class Syntax hints:   wcel 1463  wrex 2392   wss 3039  cpw 3478  cuni 3704  ciun 3781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-in 3045  df-ss 3052  df-pw 3480  df-uni 3705  df-iun 3783 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator