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Theorem iotassuni 5565
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 5561 . . 3  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 eqimss 3556 . . 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 5564 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
5 0ss 3814 . . 3  |-  (/)  C_  U. {
x  |  ph }
64, 5syl6eqss 3554 . 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 1379   E!weu 2275   {cab 2452    C_ wss 3476   (/)c0 3785   U.cuni 4245   iotacio 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030  df-uni 4246  df-iota 5549
This theorem is referenced by:  bj-nuliotaALT  33668
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