Theorem iunconst 2562

Description: Indexed union of a constant class, i.e. where B does not depend on x.

Assertion

Ref	Expression
iunconst	|- (A =/= (/) -> U_x e. A B = B)

Distinct variable groups:   x,A   x,B

Proof of Theorem iunconst

Step	Hyp	Ref	Expression
1		ne0 2278	. . . 4 |- (A =/= (/) <-> E.x x e. A)
2		ibar 641	. . . 4 |- (E.x x e. A -> (y e. B <-> (E.x x e. A /\ y e. B)))
3	1, 2	sylbi 199	. . 3 |- (A =/= (/) -> (y e. B <-> (E.x x e. A /\ y e. B)))
4		eliun 2560	. . . 4 |- (y e. U_x e. A B <-> E.x e. A y e. B)
5		df-rex 1642	. . . 4 |- (E.x e. A y e. B <-> E.x(x e. A /\ y e. B))
6		19.41v 1300	. . . 4 |- (E.x(x e. A /\ y e. B) <-> (E.x x e. A /\ y e. B))
7	4, 5, 6	3bitr 177	. . 3 |- (y e. U_x e. A B <-> (E.x x e. A /\ y e. B))
8	3, 7	syl6rbbr 537	. 2 |- (A =/= (/) -> (y e. U_x e. A B <-> y e. B))
9	8	eqrdv 1466	1 |- (A =/= (/) -> U_x e. A B = B)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577  E.wrex 1638  (/)c0 2270  U_ciun 2556

This theorem is referenced by:  abianfplem 3946  oe1m 4163  oarec 4180  oelim2 4206

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-nul 2271  df-iun 2558

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