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Theorem iotassuni 5235
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 5231 . . 3  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 eqimss 3230 . . 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 3483 . . 3  |-  (/)  C_  U. {
x  |  ph }
5 iotanul 5234 . . . 4  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
65sseq1d 3205 . . 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 1623   E!weu 2143   {cab 2269    C_ wss 3152   (/)c0 3455   U.cuni 3827   iotacio 5217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219
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