Theorem iunn0m 4026

Description: There is an inhabited class in an indexed collection B ( x ) iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)

Assertion

Ref	Expression
iunn0m	|- ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B )

Distinct variable groups:   x, y, A   y, B

Allowed substitution hint:   B( x)

Proof of Theorem iunn0m

Step	Hyp	Ref	Expression
1		rexcom4 2823	. 2 |- ( E. x e. A E. y y e. B <-> E. y E. x e. A y e. B )
2		eliun 3969	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
3	2	exbii 1651	. 2 |- ( E. y y e. U_ x e. A B <-> E. y E. x e. A y e. B )
4	1, 3	bitr4i 187	1 |- ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B )

Syntax hints:   <-> wb 105   E.wex 1538   e. wcel 2200   E.wrex 2509   U_ciun 3965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-iun 3967

This theorem is referenced by: (None)