Theorem iunn0m 3738

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 2622	. 2 |- ( E. x e. A E. y y e. B <-> E. y E. x e. A y e. B )
2		eliun 3682	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
3	2	exbii 1536	. 2 |- ( E. y y e. U_ x e. A B <-> E. y E. x e. A y e. B )
4	1, 3	bitr4i 185	1 |- ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B )

Syntax hints:   <-> wb 103   E.wex 1421   e. wcel 1433   E.wrex 2349   U_ciun 3678

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-iun 3680

This theorem is referenced by: (None)