Theorem iun0 3938

Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iun0	|- U_ x e. A (/) = (/)

Proof of Theorem iun0

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3424	. . . . . 6 |- -. y e. (/)
2	1	a1i 9	. . . . 5 |- ( x e. A -> -. y e. (/) )
3	2	nrex 2567	. . . 4 |- -. E. x e. A y e. (/)
4		eliun 3886	. . . 4 |- ( y e. U_ x e. A (/) <-> E. x e. A y e. (/) )
5	3, 4	mtbir 671	. . 3 |- -. y e. U_ x e. A (/)
6	5, 1	2false 701	. 2 |- ( y e. U_ x e. A (/) <-> y e. (/) )
7	6	eqriv 2172	1 |- U_ x e. A (/) = (/)

Syntax hints:   -. wn 3   = wceq 1353   e. wcel 2146   E.wrex 2454   (/)c0 3420   U_ciun 3882

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-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-dif 3129  df-nul 3421  df-iun 3884

This theorem is referenced by: (None)