Theorem riotassuni 6648

Description: The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
riotassuni	|- ( iota_ x e. A ph ) C_ ( ~P U. A u. U. A )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem riotassuni

Step	Hyp	Ref	Expression
1		riotauni 6617	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) = U. { x e. A | ph } )
2		ssrab2 3687	. . . . 5 |- { x e. A | ph } C_ A
3	2	unissi 4461	. . . 4 |- U. { x e. A | ph } C_ U. A
4		ssun2 3777	. . . 4 |- U. A C_ ( ~P U. A u. U. A )
5	3, 4	sstri 3612	. . 3 |- U. { x e. A | ph } C_ ( ~P U. A u. U. A )
6	1, 5	syl6eqss 3655	. 2 |- ( E! x e. A ph -> ( iota_ x e. A ph ) C_ ( ~P U. A u. U. A ) )
7		riotaund 6647	. . 3 |- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) )
8		0ss 3972	. . 3 |- (/) C_ ( ~P U. A u. U. A )
9	7, 8	syl6eqss 3655	. 2 |- ( -. E! x e. A ph -> ( iota_ x e. A ph ) C_ ( ~P U. A u. U. A ) )
10	6, 9	pm2.61i 176	1 |- ( iota_ x e. A ph ) C_ ( ~P U. A u. U. A )

Syntax hints:   -. wn 3   E!wreu 2914   {crab 2916   u. cun 3572   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   iota_crio 6610

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-reu 2919  df-rab 2921  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  df-riota 6611

This theorem is referenced by: (None)