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Theorem iotassuni 5550
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 5546 . . 3  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 eqimss 3541 . . 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 5549 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
5 0ss 3813 . . 3  |-  (/)  C_  U. {
x  |  ph }
64, 5syl6eqss 3539 . 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 1398   E!weu 2284   {cab 2439    C_ wss 3461   (/)c0 3783   U.cuni 4235   iotacio 5532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019  df-uni 4236  df-iota 5534
This theorem is referenced by:  bj-nuliotaALT  35007
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