Theorem fine 10442

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-17 973	. . . 4 |- (a e. A -> A.x a e. A)
2		snex 2756	. . . . . . 7 |- {a} e. V
3		sseq1 2085	. . . . . . . . 9 |- (y = {a} -> (y (_ A <-> {a} (_ A))
4		eleq1 1537	. . . . . . . . 9 |- (y = {a} -> (y e. Fin <-> {a} e. Fin))
5		inteq 2540	. . . . . . . . . 10 |- (y = {a} -> |^|y = |^|{a})
6	5	eqeq2d 1489	. . . . . . . . 9 |- (y = {a} -> (x = |^|y <-> x = |^|{a}))
7	3, 4, 6	3anbi123d 895	. . . . . . . 8 |- (y = {a} -> ((y (_ A /\ y e. Fin /\ x = |^|y) <-> ({a} (_ A /\ {a} e. Fin /\ x = |^|{a})))
8	7	cla4egv 1866	. . . . . . 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 2470	. . . . . . 7 |- (a e. A -> {a} (_ A)
11	10	adantl 390	. . . . . 6 |- ((x = a /\ a e. A) -> {a} (_ A)
12		snfi 4438	. . . . . . 7 |- {a} e. Fin
13	12	a1i 8	. . . . . 6 |- ((x = a /\ a e. A) -> {a} e. Fin)
14		visset 1816	. . . . . . . . . . 11 |- a e. V
15	14	intsn 2568	. . . . . . . . . 10 |- |^|{a} = a
16	15	eqcomi 1482	. . . . . . . . 9 |- a = |^|{a}
17	16	eqeq2i 1488	. . . . . . . 8 |- (x = a <-> x = |^|{a})
18	17	biimp 151	. . . . . . 7 |- (x = a -> x = |^|{a})
19	18	adantr 391	. . . . . 6 |- ((x = a /\ a e. A) -> x = |^|{a})
20	9, 11, 13, 19	syl3anc 860	. . . . 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 1162	. . 3 |- (a e. A -> E.xE.y(y (_ A /\ y e. Fin /\ x = |^|y))
23	22	19.23aiv 1297	. 2 |- (E.a a e. A -> E.xE.y(y (_ A /\ y e. Fin /\ x = |^|y))
24		ne0 2292	. 2 |- (A =/= (/) <-> E.a a e. A)
25		abn0 2294	. 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 777   = wceq 958   e. wcel 960  E.wex 982  {cab 1466   =/= wne 1588  Vcvv 1814   (_ wss 2050  (/)c0 2283  {csn 2413  |^|cint 2537  Fincfn 4373

This theorem is referenced by:  fine2 10471  fgsb 10555

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-1o 4139  df-en 4374  df-fin 4377

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