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

Theorem iotassuni 5492
Description: The  iota class is a subset of the union of all elements satisfying  ph. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
iotassuni  |-  ( iota
x ph )  C_  U. {
x  |  ph }

Proof of Theorem iotassuni
StepHypRef Expression
1 iotauni 5488 . . 3  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 eqimss 3503 . . 3  |-  ( ( iota x ph )  =  U. { x  | 
ph }  ->  ( iota x ph )  C_  U. { x  |  ph } )
31, 2syl 16 . 2  |-  ( E! x ph  ->  ( iota x ph )  C_  U. { x  |  ph } )
4 iotanul 5491 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
5 0ss 3761 . . 3  |-  (/)  C_  U. {
x  |  ph }
64, 5syl6eqss 3501 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  C_ 
U. { x  | 
ph } )
73, 6pm2.61i 164 1  |-  ( iota
x ph )  C_  U. {
x  |  ph }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370   E!weu 2260   {cab 2436    C_ wss 3423   (/)c0 3732   U.cuni 4186   iotacio 5474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-sn 3973  df-pr 3975  df-uni 4187  df-iota 5476
This theorem is referenced by:  bj-nuliotaALT  32819
  Copyright terms: Public domain W3C validator