Theorem iun0 3309

Description: An indexed union of the empty set is empty. (The proof was shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iun0	|- U_x e. A (/) = (/)

Proof of Theorem iun0

Step	Hyp	Ref	Expression
1		noel 2879	. . . . . 6 |- -. y e. (/)
2	1	a1i 8	. . . . 5 |- (x e. A -> -. y e. (/))
3	2	nrex 2192	. . . 4 |- -. E.x e. A y e. (/)
4		eliun 3259	. . . 4 |- (y e. U_x e. A (/) <-> E.x e. A y e. (/))
5	3, 4	mtbir 209	. . 3 |- -. y e. U_x e. A (/)
6	5, 1	2false 787	. 2 |- (y e. U_x e. A (/) <-> y e. (/))
7	6	eqriv 1881	1 |- U_x e. A (/) = (/)

Syntax hints:  -. wn 2   = wceq 1298   e. wcel 1300  E.wrex 2106  (/)c0 2875  U_ciun 3255

This theorem is referenced by:  dfco2aOLD 4395  om0r 5221  kmlem11 5937

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-nul 2876  df-iun 3257

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