Theorem exss 4710

Description: Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)

Assertion

Ref	Expression
exss	|- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) )

Distinct variable groups:   x, y, A   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem exss

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 2823	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	neeq1i 2752	. . 3 |- ( { x e. A | ph } =/= (/) <-> { x | ( x e. A /\ ph ) } =/= (/) )
3		rabn0 3805	. . 3 |- ( { x e. A | ph } =/= (/) <-> E. x e. A ph )
4		n0 3794	. . 3 |- ( { x | ( x e. A /\ ph ) } =/= (/) <-> E. z z e. { x | ( x e. A /\ ph ) } )
5	2, 3, 4	3bitr3i 275	. 2 |- ( E. x e. A ph <-> E. z z e. { x | ( x e. A /\ ph ) } )
6		vex 3116	. . . . . 6 |- z e. _V
7	6	snss 4151	. . . . 5 |- ( z e. { x | ( x e. A /\ ph ) } <-> { z } C_ { x | ( x e. A /\ ph ) } )
8		ssab2 3584	. . . . . 6 |- { x | ( x e. A /\ ph ) } C_ A
9		sstr2 3511	. . . . . 6 |- ( { z } C_ { x | ( x e. A /\ ph ) } -> ( { x | ( x e. A /\ ph ) } C_ A -> { z } C_ A ) )
10	8, 9	mpi 17	. . . . 5 |- ( { z } C_ { x | ( x e. A /\ ph ) } -> { z } C_ A )
11	7, 10	sylbi 195	. . . 4 |- ( z e. { x | ( x e. A /\ ph ) } -> { z } C_ A )
12		simpr 461	. . . . . . . 8 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) -> [ z / x ] ph )
13		equsb1 2080	. . . . . . . . 9 |- [ z / x ] x = z
14		elsn 4041	. . . . . . . . . 10 |- ( x e. { z } <-> x = z )
15	14	sbbii 1718	. . . . . . . . 9 |- ( [ z / x ] x e. { z } <-> [ z / x ] x = z )
16	13, 15	mpbir 209	. . . . . . . 8 |- [ z / x ] x e. { z }
17	12, 16	jctil 537	. . . . . . 7 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) -> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
18		df-clab 2453	. . . . . . . 8 |- ( z e. { x | ( x e. A /\ ph ) } <-> [ z / x ] ( x e. A /\ ph ) )
19		sban 2114	. . . . . . . 8 |- ( [ z / x ] ( x e. A /\ ph ) <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
20	18, 19	bitri 249	. . . . . . 7 |- ( z e. { x | ( x e. A /\ ph ) } <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
21		df-rab 2823	. . . . . . . . 9 |- { x e. { z } | ph } = { x | ( x e. { z } /\ ph ) }
22	21	eleq2i 2545	. . . . . . . 8 |- ( z e. { x e. { z } | ph } <-> z e. { x | ( x e. { z } /\ ph ) } )
23		df-clab 2453	. . . . . . . . 9 |- ( z e. { x | ( x e. { z } /\ ph ) } <-> [ z / x ] ( x e. { z } /\ ph ) )
24		sban 2114	. . . . . . . . 9 |- ( [ z / x ] ( x e. { z } /\ ph ) <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
25	23, 24	bitri 249	. . . . . . . 8 |- ( z e. { x | ( x e. { z } /\ ph ) } <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
26	22, 25	bitri 249	. . . . . . 7 |- ( z e. { x e. { z } | ph } <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
27	17, 20, 26	3imtr4i 266	. . . . . 6 |- ( z e. { x | ( x e. A /\ ph ) } -> z e. { x e. { z } | ph } )
28		ne0i 3791	. . . . . 6 |- ( z e. { x e. { z } | ph } -> { x e. { z } | ph } =/= (/) )
29	27, 28	syl 16	. . . . 5 |- ( z e. { x | ( x e. A /\ ph ) } -> { x e. { z } | ph } =/= (/) )
30		rabn0 3805	. . . . 5 |- ( { x e. { z } | ph } =/= (/) <-> E. x e. { z } ph )
31	29, 30	sylib 196	. . . 4 |- ( z e. { x | ( x e. A /\ ph ) } -> E. x e. { z } ph )
32		snex 4688	. . . . 5 |- { z } e. _V
33		sseq1 3525	. . . . . 6 |- ( y = { z } -> ( y C_ A <-> { z } C_ A ) )
34		rexeq 3059	. . . . . 6 |- ( y = { z } -> ( E. x e. y ph <-> E. x e. { z } ph ) )
35	33, 34	anbi12d 710	. . . . 5 |- ( y = { z } -> ( ( y C_ A /\ E. x e. y ph ) <-> ( { z } C_ A /\ E. x e. { z } ph ) ) )
36	32, 35	spcev 3205	. . . 4 |- ( ( { z } C_ A /\ E. x e. { z } ph ) -> E. y ( y C_ A /\ E. x e. y ph ) )
37	11, 31, 36	syl2anc 661	. . 3 |- ( z e. { x | ( x e. A /\ ph ) } -> E. y ( y C_ A /\ E. x e. y ph ) )
38	37	exlimiv 1698	. 2 |- ( E. z z e. { x | ( x e. A /\ ph ) } -> E. y ( y C_ A /\ E. x e. y ph ) )
39	5, 38	sylbi 195	1 |- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   E.wex 1596   [wsb 1711   e. wcel 1767   {cab 2452   =/= wne 2662   E.wrex 2815   {crab 2818   C_ wss 3476   (/)c0 3785   {csn 4027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030

This theorem is referenced by: (None)