Theorem iun0 3713

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 3228	. . . . . 6 |- -. y e. (/)
2	1	a1i 9	. . . . 5 |- ( x e. A -> -. y e. (/) )
3	2	nrex 2411	. . . 4 |- -. E. x e. A y e. (/)
4		eliun 3661	. . . 4 |- ( y e. U_ x e. A (/) <-> E. x e. A y e. (/) )
5	3, 4	mtbir 596	. . 3 |- -. y e. U_ x e. A (/)
6	5, 1	2false 617	. 2 |- ( y e. U_ x e. A (/) <-> y e. (/) )
7	6	eqriv 2037	1 |- U_ x e. A (/) = (/)

Syntax hints:   -. wn 3   = wceq 1243   e. wcel 1393   E.wrex 2307   (/)c0 3224   U_ciun 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-nul 3225  df-iun 3659

This theorem is referenced by: (None)