Theorem fine 10213

Description: Condition required for a nonempty finite intersection. (Contributed by FL, 2-Sep-2008.)

Assertion

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

Distinct variable group:   x,A,y

Proof of Theorem fine

Step	Hyp	Ref	Expression
1		ax-17 1317	. . . 4 |- (a e. A -> A.x a e. A)
2		snssi 3129	. . . . . . 7 |- (a e. A -> {a} C_ A)
3	2	adantl 424	. . . . . 6 |- ((x = a /\ a e. A) -> {a} C_ A)
4		snfi 5491	. . . . . . 7 |- {a} e. Fin
5	4	a1i 8	. . . . . 6 |- ((x = a /\ a e. A) -> {a} e. Fin)
6		visset 2295	. . . . . . . . . . 11 |- a e. _V
7	6	intsn 3252	. . . . . . . . . 10 |- |^|{a} = a
8	7	eqcomi 1888	. . . . . . . . 9 |- a = |^|{a}
9	8	eqeq2i 1894	. . . . . . . 8 |- (x = a <-> x = |^|{a})
10	9	biimpi 168	. . . . . . 7 |- (x = a -> x = |^|{a})
11	10	adantr 425	. . . . . 6 |- ((x = a /\ a e. A) -> x = |^|{a})
12		snex 3492	. . . . . . 7 |- {a} e. _V
13		sseq1 2637	. . . . . . . . 9 |- (y = {a} -> (y C_ A <-> {a} C_ A))
14		eleq1 1957	. . . . . . . . 9 |- (y = {a} -> (y e. Fin <-> {a} e. Fin))
15		inteq 3217	. . . . . . . . . 10 |- (y = {a} -> |^|y = |^|{a})
16	15	eqeq2d 1895	. . . . . . . . 9 |- (y = {a} -> (x = |^|y <-> x = |^|{a}))
17	13, 14, 16	3anbi123d 1168	. . . . . . . 8 |- (y = {a} -> ((y C_ A /\ y e. Fin /\ x = |^|y) <-> ({a} C_ A /\ {a} e. Fin /\ x = |^|{a})))
18	17	cla4egv 2365	. . . . . . 7 |- ({a} e. _V -> (({a} C_ A /\ {a} e. Fin /\ x = |^|{a}) -> E.y(y C_ A /\ y e. Fin /\ x = |^|y)))
19	12, 18	ax-mp 7	. . . . . 6 |- (({a} C_ A /\ {a} e. Fin /\ x = |^|{a}) -> E.y(y C_ A /\ y e. Fin /\ x = |^|y))
20	3, 5, 11, 19	syl111anc 1100	. . . . 5 |- ((x = a /\ a e. A) -> E.y(y C_ A /\ y e. Fin /\ x = |^|y))
21	20	ex 402	. . . 4 |- (x = a -> (a e. A -> E.y(y C_ A /\ y e. Fin /\ x = |^|y)))
22	1, 21	a4ime 1521	. . 3 |- (a e. A -> E.xE.y(y C_ A /\ y e. Fin /\ x = |^|y))
23	22	19.23aiv 1674	. 2 |- (E.a a e. A -> E.xE.y(y C_ A /\ y e. Fin /\ x = |^|y))
24		n0 2884	. 2 |- (A =/= (/) <-> E.a a e. A)
25		abn0 2892	. 2 |- ({x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} =/= (/) <-> E.xE.y(y C_ A /\ y e. Fin /\ x = |^|y))
26	23, 24, 25	3imtr4i 236	1 |- (A =/= (/) -> {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   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {csn 3044  |^|cint 3214  Fincfn 5426

This theorem is referenced by:  fine2 10214  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  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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