Theorem iun0 3929

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 3418	. . . . . 6 |- -. y e. (/)
2	1	a1i 9	. . . . 5 |- ( x e. A -> -. y e. (/) )
3	2	nrex 2562	. . . 4 |- -. E. x e. A y e. (/)
4		eliun 3877	. . . 4 |- ( y e. U_ x e. A (/) <-> E. x e. A y e. (/) )
5	3, 4	mtbir 666	. . 3 |- -. y e. U_ x e. A (/)
6	5, 1	2false 696	. 2 |- ( y e. U_ x e. A (/) <-> y e. (/) )
7	6	eqriv 2167	1 |- U_ x e. A (/) = (/)

Syntax hints:   -. wn 3   = wceq 1348   e. wcel 2141   E.wrex 2449   (/)c0 3414   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-in1 609  ax-in2 610  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-fal 1354  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-dif 3123  df-nul 3415  df-iun 3875

This theorem is referenced by: (None)