Theorem iun0 4022

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 3554	. . . . . 6 |- -. y e. (/)
2	1	a1i 10	. . . . 5 |- (x e. A -> -. y e. (/))
3	2	nrex 2716	. . . 4 |- -. E.x e. A y e. (/)
4		eliun 3973	. . . 4 |- (y e. U_x e. A (/) <-> E.x e. A y e. (/))
5	3, 4	mtbir 290	. . 3 |- -. y e. U_x e. A (/)
6	5, 1	2false 339	. 2 |- (y e. U_x e. A (/) <-> y e. (/))
7	6	eqriv 2350	1 |- U_x e. A (/) = (/)

Syntax hints:  -. wn 3   = wceq 1642   e. wcel 1710  E.wrex 2615  (/)c0 3550  U_ciun 3969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-iun 3971

This theorem is referenced by:  iununi  4050