Theorem iunconst 2576

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 2292	. . . 4 |- (A =/= (/) <-> E.x x e. A)
2		ibar 645	. . . 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 2574	. . . 4 |- (y e. U_x e. A B <-> E.x e. A y e. B)
5		df-rex 1653	. . . 4 |- (E.x e. A y e. B <-> E.x(x e. A /\ y e. B))
6		19.41v 1307	. . . 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 541	. 2 |- (A =/= (/) -> (y e. U_x e. A B <-> y e. B))
9	8	eqrdv 1476	1 |- (A =/= (/) -> U_x e. A B = B)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982   =/= wne 1588  E.wrex 1649  (/)c0 2283  U_ciun 2570

This theorem is referenced by:  abianfplem 3967  oe1m 4185  oarec 4202  oelim2 4228

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-nul 2284  df-iun 2572

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