Theorem iunn0m 4002

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 2800	. 2 |- ( E. x e. A E. y y e. B <-> E. y E. x e. A y e. B )
2		eliun 3945	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
3	2	exbii 1629	. 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 1516   e. wcel 2178   E.wrex 2487   U_ciun 3941

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-iun 3943

This theorem is referenced by: (None)