Theorem abfi 10215

Description: Any element of A is the intersection of a finite subclass of A. (Contributed by FL, 2-Sep-2008.)

Assertion

Ref	Expression
abfi	|- A C_ {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)}

Distinct variable group:   x,A,y

Proof of Theorem abfi

Step	Hyp	Ref	Expression
1		ssab 2677	. 2 |- (A C_ {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} <-> A.x(x e. A -> E.y(y C_ A /\ y e. Fin /\ x = |^|y)))
2		snfi 5491	. . . 4 |- {x} e. Fin
3		visset 2295	. . . . . 6 |- x e. _V
4	3	intsn 3252	. . . . 5 |- |^|{x} = x
5	4	eqcomi 1888	. . . 4 |- x = |^|{x}
6		eleq1 1957	. . . . . . 7 |- (y = {x} -> (y e. Fin <-> {x} e. Fin))
7		inteq 3217	. . . . . . . 8 |- (y = {x} -> |^|y = |^|{x})
8	7	eqeq2d 1895	. . . . . . 7 |- (y = {x} -> (x = |^|y <-> x = |^|{x}))
9	6, 8	anbi12d 690	. . . . . 6 |- (y = {x} -> ((y e. Fin /\ x = |^|y) <-> ({x} e. Fin /\ x = |^|{x})))
10	9	rcla4ev 2381	. . . . 5 |- (({x} e. ~PA /\ ({x} e. Fin /\ x = |^|{x})) -> E.y e. ~P A(y e. Fin /\ x = |^|y))
11	10	ex 402	. . . 4 |- ({x} e. ~PA -> (({x} e. Fin /\ x = |^|{x}) -> E.y e. ~P A(y e. Fin /\ x = |^|y)))
12	2, 5, 11	mp2ani 764	. . 3 |- ({x} e. ~PA -> E.y e. ~P A(y e. Fin /\ x = |^|y))
13	3	snelpw 3501	. . 3 |- (x e. A <-> {x} e. ~PA)
14		df-rex 2110	. . . 4 |- (E.y e. ~P A(y e. Fin /\ x = |^|y) <-> E.y(y e. ~PA /\ (y e. Fin /\ x = |^|y)))
15		visset 2295	. . . . . . . 8 |- y e. _V
16	15	elpw 3037	. . . . . . 7 |- (y e. ~PA <-> y C_ A)
17	16	anbi1i 539	. . . . . 6 |- ((y e. ~PA /\ (y e. Fin /\ x = |^|y)) <-> (y C_ A /\ (y e. Fin /\ x = |^|y)))
18		3anass 862	. . . . . 6 |- ((y C_ A /\ y e. Fin /\ x = |^|y) <-> (y C_ A /\ (y e. Fin /\ x = |^|y)))
19	17, 18	bitr4i 193	. . . . 5 |- ((y e. ~PA /\ (y e. Fin /\ x = |^|y)) <-> (y C_ A /\ y e. Fin /\ x = |^|y))
20	19	exbii 1398	. . . 4 |- (E.y(y e. ~PA /\ (y e. Fin /\ x = |^|y)) <-> E.y(y C_ A /\ y e. Fin /\ x = |^|y))
21	14, 20	bitr2i 191	. . 3 |- (E.y(y C_ A /\ y e. Fin /\ x = |^|y) <-> E.y e. ~P A(y e. Fin /\ x = |^|y))
22	12, 13, 21	3imtr4i 236	. 2 |- (x e. A -> E.y(y C_ A /\ y e. Fin /\ x = |^|y))
23	1, 22	mpgbir 1334	1 |- A C_ {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)}

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  E.wrex 2106   C_ wss 2593  ~Pcpw 3032  {csn 3044  |^|cint 3214  Fincfn 5426

This theorem is referenced by:  abfi2 10216  efilcp 14922

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-13 1311  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  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-1o 5177  df-en 5427  df-fin 5430

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