Theorem fiiu 14344

Description: If A is the intersection of a finite set of elements of B then A C_ U.B.

Assertion

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

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

Proof of Theorem fiiu

Step	Hyp	Ref	Expression
1		nvel 3450	. . . 4 |- -. _V e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)}
2		eleq1 1957	. . . . 5 |- (A = _V -> (A e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)} <-> _V e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)}))
3	2	biimpcd 172	. . . 4 |- (A e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)} -> (A = _V -> _V e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)}))
4	1, 3	mtoi 122	. . 3 |- (A e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)} -> -. A = _V)
5		spfi 10217	. . . 4 |- (A e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)} -> (A e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)} <-> E.y(y C_ B /\ y e. Fin /\ A = |^|y)))
6		eqeq1 1890	. . . . . . . . . . . 12 |- (A = |^|y -> (A = _V <-> |^|y = _V))
7	6	notbid 673	. . . . . . . . . . 11 |- (A = |^|y -> (-. A = _V <-> -. |^|y = _V))
8	7	biimpd 170	. . . . . . . . . 10 |- (A = |^|y -> (-. A = _V -> -. |^|y = _V))
9		int0 3230	. . . . . . . . . . 11 |- |^|(/) = _V
10		eqeq2 1893	. . . . . . . . . . . . . 14 |- (|^|(/) = _V -> (|^|y = |^|(/) <-> |^|y = _V))
11	10	bicomd 580	. . . . . . . . . . . . 13 |- (|^|(/) = _V -> (|^|y = _V <-> |^|y = |^|(/)))
12	11	notbid 673	. . . . . . . . . . . 12 |- (|^|(/) = _V -> (-. |^|y = _V <-> -. |^|y = |^|(/)))
13		inteq 3217	. . . . . . . . . . . . . 14 |- (y = (/) -> |^|y = |^|(/))
14	13	con3i 114	. . . . . . . . . . . . 13 |- (-. |^|y = |^|(/) -> -. y = (/))
15		sseq1 2637	. . . . . . . . . . . . . . . . . 18 |- (|^|y = A -> (|^|y C_ U.B <-> A C_ U.B))
16	15	biimpd 170	. . . . . . . . . . . . . . . . 17 |- (|^|y = A -> (|^|y C_ U.B -> A C_ U.B))
17	16	eqcoms 1887	. . . . . . . . . . . . . . . 16 |- (A = |^|y -> (|^|y C_ U.B -> A C_ U.B))
18		intssuni2 3243	. . . . . . . . . . . . . . . . . . 19 |- ((y C_ B /\ y =/= (/)) -> |^|y C_ U.B)
19		df-ne 2019	. . . . . . . . . . . . . . . . . . 19 |- (y =/= (/) <-> -. y = (/))
20	18, 19	sylan2br 502	. . . . . . . . . . . . . . . . . 18 |- ((y C_ B /\ -. y = (/)) -> |^|y C_ U.B)
21	20	ancoms 484	. . . . . . . . . . . . . . . . 17 |- ((-. y = (/) /\ y C_ B) -> |^|y C_ U.B)
22	21	3adant2 895	. . . . . . . . . . . . . . . 16 |- ((-. y = (/) /\ y e. Fin /\ y C_ B) -> |^|y C_ U.B)
23	17, 22	syl5com 63	. . . . . . . . . . . . . . 15 |- ((-. y = (/) /\ y e. Fin /\ y C_ B) -> (A = |^|y -> A C_ U.B))
24	23	3exp 1066	. . . . . . . . . . . . . 14 |- (-. y = (/) -> (y e. Fin -> (y C_ B -> (A = |^|y -> A C_ U.B))))
25	24	com24 41	. . . . . . . . . . . . 13 |- (-. y = (/) -> (A = |^|y -> (y C_ B -> (y e. Fin -> A C_ U.B))))
26	14, 25	syl 12	. . . . . . . . . . . 12 |- (-. |^|y = |^|(/) -> (A = |^|y -> (y C_ B -> (y e. Fin -> A C_ U.B))))
27	12, 26	syl6bi 231	. . . . . . . . . . 11 |- (|^|(/) = _V -> (-. |^|y = _V -> (A = |^|y -> (y C_ B -> (y e. Fin -> A C_ U.B)))))
28	9, 27	ax-mp 7	. . . . . . . . . 10 |- (-. |^|y = _V -> (A = |^|y -> (y C_ B -> (y e. Fin -> A C_ U.B))))
29	8, 28	syl6com 64	. . . . . . . . 9 |- (-. A = _V -> (A = |^|y -> (A = |^|y -> (y C_ B -> (y e. Fin -> A C_ U.B)))))
30	29	com13 37	. . . . . . . 8 |- (A = |^|y -> (A = |^|y -> (-. A = _V -> (y C_ B -> (y e. Fin -> A C_ U.B)))))
31	30	pm2.43i 78	. . . . . . 7 |- (A = |^|y -> (-. A = _V -> (y C_ B -> (y e. Fin -> A C_ U.B))))
32	31	com4t 44	. . . . . 6 |- (y C_ B -> (y e. Fin -> (A = |^|y -> (-. A = _V -> A C_ U.B))))
33	32	3imp 1061	. . . . 5 |- ((y C_ B /\ y e. Fin /\ A = |^|y) -> (-. A = _V -> A C_ U.B))
34	33	19.23aiv 1674	. . . 4 |- (E.y(y C_ B /\ y e. Fin /\ A = |^|y) -> (-. A = _V -> A C_ U.B))
35	5, 34	syl6bi 231	. . 3 |- (A e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)} -> (A e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)} -> (-. A = _V -> A C_ U.B)))
36	4, 35	mpid 58	. 2 |- (A e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)} -> (A e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)} -> A C_ U.B))
37	36	pm2.43i 78	1 |- (A e. {x | E.y(y C_ B /\ y e. Fin /\ x = |^|y)} -> A C_ U.B)

Syntax hints:  -. wn 2   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  U.cuni 3177  |^|cint 3214  Fincfn 5426

This theorem is referenced by:  fgsb 14921

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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-uni 3178  df-int 3215

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