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Metamath Proof Explorer

Theorem iunn0

Description: There is a nonempty class in an indexed collection B ( x ) iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)

Ref Expression
Assertion iunn0 
|- ( E. x e. A B =/= (/) <-> U_ x e. A B =/= (/) )

Proof

Step Hyp Ref Expression
1 rexcom4 
 |-  ( E. x e. A E. y y e. B <-> E. y E. x e. A y e. B )
2 eliun 
 |-  ( y e. U_ x e. A B <-> E. x e. A y e. B )
3 2 exbii 
 |-  ( E. y y e. U_ x e. A B <-> E. y E. x e. A y e. B )
4 1 3 bitr4i 
 |-  ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B )
5 n0 
 |-  ( B =/= (/) <-> E. y y e. B )
6 5 rexbii 
 |-  ( E. x e. A B =/= (/) <-> E. x e. A E. y y e. B )
7 n0 
 |-  ( U_ x e. A B =/= (/) <-> E. y y e. U_ x e. A B )
8 4 6 7 3bitr4i 
 |-  ( E. x e. A B =/= (/) <-> U_ x e. A B =/= (/) )