Theorem iunn0m 3933

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 2753	. 2 |- ( E. x e. A E. y y e. B <-> E. y E. x e. A y e. B )
2		eliun 3877	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
3	2	exbii 1598	. 2 |- ( E. y y e. U_ x e. A B <-> E. y E. x e. A y e. B )
4	1, 3	bitr4i 186	1 |- ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B )

Syntax hints:   <-> wb 104   E.wex 1485   e. wcel 2141   E.wrex 2449   U_ciun 3873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-iun 3875

This theorem is referenced by: (None)