Theorem iotassuni 5867

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

Step	Hyp	Ref	Expression
1		iotauni 5863	. . 3 |- ( E! x ph -> ( iota x ph ) = U. { x | ph } )
2		eqimss 3657	. . 3 |- ( ( iota x ph ) = U. { x | ph } -> ( iota x ph ) C_ U. { x | ph } )
3	1, 2	syl 17	. 2 |- ( E! x ph -> ( iota x ph ) C_ U. { x | ph } )
4		iotanul 5866	. . 3 |- ( -. E! x ph -> ( iota x ph ) = (/) )
5		0ss 3972	. . 3 |- (/) C_ U. { x | ph }
6	4, 5	syl6eqss 3655	. 2 |- ( -. E! x ph -> ( iota x ph ) C_ U. { x | ph } )
7	3, 6	pm2.61i 176	1 |- ( iota x ph ) C_ U. { x | ph }

Syntax hints:   -. wn 3   = wceq 1483   E!weu 2470   {cab 2608   C_ wss 3574   (/)c0 3915   U.cuni 4436   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by:  bj-nuliotaALT  33020