Theorem iun0 4067

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 3525	. . . . . 6 |- -. y e. (/)
2	1	a1i 9	. . . . 5 |- ( x e. A -> -. y e. (/) )
3	2	nrex 2642	. . . 4 |- -. E. x e. A y e. (/)
4		eliun 4014	. . . 4 |- ( y e. U_ x e. A (/) <-> E. x e. A y e. (/) )
5	3, 4	mtbir 682	. . 3 |- -. y e. U_ x e. A (/)
6	5, 1	2false 713	. 2 |- ( y e. U_ x e. A (/) <-> y e. (/) )
7	6	eqriv 2235	1 |- U_ x e. A (/) = (/)

Syntax hints:   -. wn 3   = wceq 1402   e. wcel 2209   E.wrex 2529   (/)c0 3520   U_ciun 4010

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-nul 3521  df-iun 4012

This theorem is referenced by: (None)