Theorem iunn0m 3717

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 2577	. 2 |- ( E. x e. A E. y y e. B <-> E. y E. x e. A y e. B )
2		eliun 3661	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
3	2	exbii 1496	. 2 |- ( E. y y e. U_ x e. A B <-> E. y E. x e. A y e. B )
4	1, 3	bitr4i 176	1 |- ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B )

Syntax hints:   <-> wb 98   E.wex 1381   e. wcel 1393   E.wrex 2307   U_ciun 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-iun 3659

This theorem is referenced by: (None)