Theorem exss 3516

Description: Restricted existence in a class (even if proper) implies restricted existence in a subset.

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

Proof of Theorem exss

Step	Hyp	Ref	Expression
1		df-rab 2112	. . . 4 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2	1	neeq1i 2026	. . 3 |- ({x e. A | ph} =/= (/) <-> {x | (x e. A /\ ph)} =/= (/))
3		rabn0 2893	. . 3 |- ({x e. A | ph} =/= (/) <-> E.x e. A ph)
4		n0 2884	. . 3 |- ({x | (x e. A /\ ph)} =/= (/) <-> E.z z e. {x | (x e. A /\ ph)})
5	2, 3, 4	3bitr3i 198	. 2 |- (E.x e. A ph <-> E.z z e. {x | (x e. A /\ ph)})
6		visset 2295	. . . . . 6 |- z e. _V
7	6	snss 3122	. . . . 5 |- (z e. {x | (x e. A /\ ph)} <-> {z} C_ {x | (x e. A /\ ph)})
8		ssab2 2691	. . . . . 6 |- {x | (x e. A /\ ph)} C_ A
9		sstr2 2623	. . . . . 6 |- ({z} C_ {x | (x e. A /\ ph)} -> ({x | (x e. A /\ ph)} C_ A -> {z} C_ A))
10	8, 9	mpi 55	. . . . 5 |- ({z} C_ {x | (x e. A /\ ph)} -> {z} C_ A)
11	7, 10	sylbi 216	. . . 4 |- (z e. {x | (x e. A /\ ph)} -> {z} C_ A)
12		simpr 350	. . . . . . . 8 |- (([z / x]x e. A /\ [z / x]ph) -> [z / x]ph)
13		equsb1 1561	. . . . . . . . 9 |- [z / x]x = z
14		elsn 3058	. . . . . . . . . 10 |- (x e. {z} <-> x = z)
15	14	sbbii 1538	. . . . . . . . 9 |- ([z / x]x e. {z} <-> [z / x]x = z)
16	13, 15	mpbir 207	. . . . . . . 8 |- [z / x]x e. {z}
17	12, 16	jctil 316	. . . . . . 7 |- (([z / x]x e. A /\ [z / x]ph) -> ([z / x]x e. {z} /\ [z / x]ph))
18		df-clab 1872	. . . . . . . 8 |- (z e. {x | (x e. A /\ ph)} <-> [z / x](x e. A /\ ph))
19		sban 1607	. . . . . . . 8 |- ([z / x](x e. A /\ ph) <-> ([z / x]x e. A /\ [z / x]ph))
20	18, 19	bitri 190	. . . . . . 7 |- (z e. {x | (x e. A /\ ph)} <-> ([z / x]x e. A /\ [z / x]ph))
21		df-rab 2112	. . . . . . . . 9 |- {x e. {z} | ph} = {x | (x e. {z} /\ ph)}
22	21	eleq2i 1961	. . . . . . . 8 |- (z e. {x e. {z} | ph} <-> z e. {x | (x e. {z} /\ ph)})
23		df-clab 1872	. . . . . . . 8 |- (z e. {x | (x e. {z} /\ ph)} <-> [z / x](x e. {z} /\ ph))
24		sban 1607	. . . . . . . 8 |- ([z / x](x e. {z} /\ ph) <-> ([z / x]x e. {z} /\ [z / x]ph))
25	22, 23, 24	3bitri 194	. . . . . . 7 |- (z e. {x e. {z} | ph} <-> ([z / x]x e. {z} /\ [z / x]ph))
26	17, 20, 25	3imtr4i 236	. . . . . 6 |- (z e. {x | (x e. A /\ ph)} -> z e. {x e. {z} | ph})
27		ne0i 2881	. . . . . 6 |- (z e. {x e. {z} | ph} -> {x e. {z} | ph} =/= (/))
28	26, 27	syl 12	. . . . 5 |- (z e. {x | (x e. A /\ ph)} -> {x e. {z} | ph} =/= (/))
29		rabn0 2893	. . . . 5 |- ({x e. {z} | ph} =/= (/) <-> E.x e. {z}ph)
30	28, 29	sylib 215	. . . 4 |- (z e. {x | (x e. A /\ ph)} -> E.x e. {z}ph)
31		snex 3492	. . . . 5 |- {z} e. _V
32		sseq1 2637	. . . . . 6 |- (y = {z} -> (y C_ A <-> {z} C_ A))
33		rexeq 2267	. . . . . 6 |- (y = {z} -> (E.x e. y ph <-> E.x e. {z}ph))
34	32, 33	anbi12d 690	. . . . 5 |- (y = {z} -> ((y C_ A /\ E.x e. y ph) <-> ({z} C_ A /\ E.x e. {z}ph)))
35	31, 34	cla4ev 2371	. . . 4 |- (({z} C_ A /\ E.x e. {z}ph) -> E.y(y C_ A /\ E.x e. y ph))
36	11, 30, 35	syl11anc 524	. . 3 |- (z e. {x | (x e. A /\ ph)} -> E.y(y C_ A /\ E.x e. y ph))
37	36	19.23aiv 1674	. 2 |- (E.z z e. {x | (x e. A /\ ph)} -> E.y(y C_ A /\ E.x e. y ph))
38	5, 37	sylbi 216	1 |- (E.x e. A ph -> E.y(y C_ A /\ E.x e. y ph))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  {cab 1871   =/= wne 2017  E.wrex 2106  {crab 2108   C_ wss 2593  (/)c0 2875  {csn 3044

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049

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