Theorem iun0 3942

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 3426	. . . . . 6 |- -. y e. (/)
2	1	a1i 9	. . . . 5 |- ( x e. A -> -. y e. (/) )
3	2	nrex 2569	. . . 4 |- -. E. x e. A y e. (/)
4		eliun 3890	. . . 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 2174	1 |- U_ x e. A (/) = (/)

Syntax hints:   -. wn 3   = wceq 1353   e. wcel 2148   E.wrex 2456   (/)c0 3422   U_ciun 3886

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-nul 3423  df-iun 3888

This theorem is referenced by: (None)