Metamath Proof Explorer


Theorem infssuni

Description: If an infinite set A is included in the underlying set of a finite cover B , then there exists a set of the cover that contains an infinite number of element of A . (Contributed by FL, 2-Aug-2009)

Ref Expression
Assertion infssuni ¬AFinBFinABxB¬AxFin

Proof

Step Hyp Ref Expression
1 dfral2 xBAxFin¬xB¬AxFin
2 iunfi BFinxBAxFinxBAxFin
3 iunin2 xBAx=AxBx
4 3 eleq1i xBAxFinAxBxFin
5 uniiun B=xBx
6 5 eqcomi xBx=B
7 6 ineq2i AxBx=AB
8 7 eleq1i AxBxFinABFin
9 df-ss ABAB=A
10 eleq1 AB=AABFinAFin
11 pm2.24 AFin¬AFinxB¬AxFin
12 10 11 syl6bi AB=AABFin¬AFinxB¬AxFin
13 9 12 sylbi ABABFin¬AFinxB¬AxFin
14 13 com12 ABFinAB¬AFinxB¬AxFin
15 8 14 sylbi AxBxFinAB¬AFinxB¬AxFin
16 4 15 sylbi xBAxFinAB¬AFinxB¬AxFin
17 2 16 syl BFinxBAxFinAB¬AFinxB¬AxFin
18 17 ex BFinxBAxFinAB¬AFinxB¬AxFin
19 18 com24 BFin¬AFinABxBAxFinxB¬AxFin
20 19 3imp21 ¬AFinBFinABxBAxFinxB¬AxFin
21 1 20 biimtrrid ¬AFinBFinAB¬xB¬AxFinxB¬AxFin
22 21 pm2.18d ¬AFinBFinABxB¬AxFin