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Theorem iotassuni 5251
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 5247 . . 3  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 eqimss 3243 . . 3  |-  ( ( iota x ph )  =  U. { x  | 
ph }  ->  ( iota x ph )  C_  U. { x  |  ph } )
31, 2syl 15 . 2  |-  ( E! x ph  ->  ( iota x ph )  C_  U. { x  |  ph } )
4 0ss 3496 . . 3  |-  (/)  C_  U. {
x  |  ph }
5 iotanul 5250 . . . 4  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
65sseq1d 3218 . . 3  |-  ( -.  E! x ph  ->  ( ( iota x ph )  C_  U. { x  |  ph }  <->  (/)  C_  U. {
x  |  ph }
) )
74, 6mpbiri 224 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  C_ 
U. { x  | 
ph } )
83, 7pm2.61i 156 1  |-  ( iota
x ph )  C_  U. {
x  |  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632   E!weu 2156   {cab 2282    C_ wss 3165   (/)c0 3468   U.cuni 3843   iotacio 5233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235
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