Theorem fisub 10576

Description: If a set has the finite intersection property, its subsets have also this property.

Hypotheses

Ref	Expression
fisubNEW.1	|- B = {z | E.y(y (_ A /\ y e. Fin /\ z = |^|y)}
fisubNEW.2	|- D = {z | E.y(y (_ C /\ y e. Fin /\ z = |^|y)}

Assertion

Ref	Expression
fisub	|- (C (_ A -> (-. (/) e. B -> -. (/) e. D))

Distinct variable groups:   y,A,z   y,B   y,C,z

Proof of Theorem fisub

Step	Hyp	Ref	Expression
1		sstr 2072	. . . . . . . . 9 |- ((y (_ C /\ C (_ A) -> y (_ A)
2		0ex 2711	. . . . . . . . . . . . 13 |- (/) e. V
3		eqeq1 1481	. . . . . . . . . . . . . . 15 |- (z = (/) -> (z = |^|y <-> (/) = |^|y))
4	3	3anbi3d 899	. . . . . . . . . . . . . 14 |- (z = (/) -> ((y (_ A /\ y e. Fin /\ z = |^|y) <-> (y (_ A /\ y e. Fin /\ (/) = |^|y)))
5	4	exbidv 1279	. . . . . . . . . . . . 13 |- (z = (/) -> (E.y(y (_ A /\ y e. Fin /\ z = |^|y) <-> E.y(y (_ A /\ y e. Fin /\ (/) = |^|y)))
6		fisubNEW.1	. . . . . . . . . . . . 13 |- B = {z | E.y(y (_ A /\ y e. Fin /\ z = |^|y)}
7	2, 5, 6	elab2 1901	. . . . . . . . . . . 12 |- ((/) e. B <-> E.y(y (_ A /\ y e. Fin /\ (/) = |^|y))
8	7	biimpr 152	. . . . . . . . . . 11 |- (E.y(y (_ A /\ y e. Fin /\ (/) = |^|y) -> (/) e. B)
9	8	19.23bi 1065	. . . . . . . . . 10 |- ((y (_ A /\ y e. Fin /\ (/) = |^|y) -> (/) e. B)
10	9	3exp 832	. . . . . . . . 9 |- (y (_ A -> (y e. Fin -> ((/) = |^|y -> (/) e. B)))
11	1, 10	syl 10	. . . . . . . 8 |- ((y (_ C /\ C (_ A) -> (y e. Fin -> ((/) = |^|y -> (/) e. B)))
12	11	expcom 374	. . . . . . 7 |- (C (_ A -> (y (_ C -> (y e. Fin -> ((/) = |^|y -> (/) e. B))))
13	12	com4l 39	. . . . . 6 |- (y (_ C -> (y e. Fin -> ((/) = |^|y -> (C (_ A -> (/) e. B))))
14	13	3imp 827	. . . . 5 |- ((y (_ C /\ y e. Fin /\ (/) = |^|y) -> (C (_ A -> (/) e. B))
15	14	19.23aiv 1295	. . . 4 |- (E.y(y (_ C /\ y e. Fin /\ (/) = |^|y) -> (C (_ A -> (/) e. B))
16	15	com12 11	. . 3 |- (C (_ A -> (E.y(y (_ C /\ y e. Fin /\ (/) = |^|y) -> (/) e. B))
17	3	3anbi3d 899	. . . . 5 |- (z = (/) -> ((y (_ C /\ y e. Fin /\ z = |^|y) <-> (y (_ C /\ y e. Fin /\ (/) = |^|y)))
18	17	exbidv 1279	. . . 4 |- (z = (/) -> (E.y(y (_ C /\ y e. Fin /\ z = |^|y) <-> E.y(y (_ C /\ y e. Fin /\ (/) = |^|y)))
19		fisubNEW.2	. . . 4 |- D = {z | E.y(y (_ C /\ y e. Fin /\ z = |^|y)}
20	2, 18, 19	elab2 1901	. . 3 |- ((/) e. D <-> E.y(y (_ C /\ y e. Fin /\ (/) = |^|y))
21	16, 20	syl5ib 206	. 2 |- (C (_ A -> ((/) e. D -> (/) e. B))
22	21	con3d 95	1 |- (C (_ A -> (-. (/) e. B -> -. (/) e. D))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980  {cab 1463   (_ wss 2047  (/)c0 2280  |^|cint 2533  Fincfn 4367

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-10 966  ax-11 967  ax-12 968  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-nul 2710

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281

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