Theorem fine 10449

Description: Condition required for a nonempty finite intersection.

Assertion

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

Distinct variable group:   x,A,y

Proof of Theorem fine

Step	Hyp	Ref	Expression
1		ax-17 971	. . . 4 |- (a e. A -> A.x a e. A)
2		snex 2750	. . . . . . 7 |- {a} e. V
3		sseq1 2082	. . . . . . . . 9 |- (y = {a} -> (y (_ A <-> {a} (_ A))
4		eleq1 1534	. . . . . . . . 9 |- (y = {a} -> (y e. Fin <-> {a} e. Fin))
5		inteq 2536	. . . . . . . . . 10 |- (y = {a} -> |^|y = |^|{a})
6	5	eqeq2d 1486	. . . . . . . . 9 |- (y = {a} -> (x = |^|y <-> x = |^|{a}))
7	3, 4, 6	3anbi123d 893	. . . . . . . 8 |- (y = {a} -> ((y (_ A /\ y e. Fin /\ x = |^|y) <-> ({a} (_ A /\ {a} e. Fin /\ x = |^|{a})))
8	7	cla4egv 1863	. . . . . . 7 |- ({a} e. V -> (({a} (_ A /\ {a} e. Fin /\ x = |^|{a}) -> E.y(y (_ A /\ y e. Fin /\ x = |^|y)))
9	2, 8	ax-mp 7	. . . . . 6 |- (({a} (_ A /\ {a} e. Fin /\ x = |^|{a}) -> E.y(y (_ A /\ y e. Fin /\ x = |^|y))
10		snssi 2466	. . . . . . 7 |- (a e. A -> {a} (_ A)
11	10	adantl 388	. . . . . 6 |- ((x = a /\ a e. A) -> {a} (_ A)
12		snfi 4432	. . . . . . 7 |- {a} e. Fin
13	12	a1i 8	. . . . . 6 |- ((x = a /\ a e. A) -> {a} e. Fin)
14		visset 1813	. . . . . . . . . . 11 |- a e. V
15	14	intsn 2564	. . . . . . . . . 10 |- |^|{a} = a
16	15	eqcomi 1479	. . . . . . . . 9 |- a = |^|{a}
17	16	eqeq2i 1485	. . . . . . . 8 |- (x = a <-> x = |^|{a})
18	17	biimp 151	. . . . . . 7 |- (x = a -> x = |^|{a})
19	18	adantr 389	. . . . . 6 |- ((x = a /\ a e. A) -> x = |^|{a})
20	9, 11, 13, 19	syl3anc 858	. . . . 5 |- ((x = a /\ a e. A) -> E.y(y (_ A /\ y e. Fin /\ x = |^|y))
21	20	ex 373	. . . 4 |- (x = a -> (a e. A -> E.y(y (_ A /\ y e. Fin /\ x = |^|y)))
22	1, 21	a4ime 1160	. . 3 |- (a e. A -> E.xE.y(y (_ A /\ y e. Fin /\ x = |^|y))
23	22	19.23aiv 1295	. 2 |- (E.a a e. A -> E.xE.y(y (_ A /\ y e. Fin /\ x = |^|y))
24		ne0 2288	. 2 |- (A =/= (/) <-> E.a a e. A)
25		abn0 2290	. 2 |- ({x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)} =/= (/) <-> E.xE.y(y (_ A /\ y e. Fin /\ x = |^|y))
26	23, 24, 25	3imtr4 219	1 |- (A =/= (/) -> {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)} =/= (/))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980  {cab 1463   =/= wne 1585  Vcvv 1811   (_ wss 2047  (/)c0 2280  {csn 2409  |^|cint 2533  Fincfn 4367

This theorem is referenced by:  fine2 10484  fgsb 10570

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-1o 4133  df-en 4368  df-fin 4371

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