Theorem unisseq 1946

Description: A set is a union of its subsets.

Assertion

Ref	Expression
unisseq	|- (A e. B -> A = U.{x e. B | x (_ A})

Distinct variable group(s):   x,A   x,B

Proof of Theorem unisseq

Step	Hyp	Ref	Expression
1		ssid 1519	. . . 4 |- A (_ A
2	1	jctr 239	. . 3 |- (A e. B -> (A e. B /\ A (_ A))
3		sseq1 1521	. . . 4 |- (x = A -> (x (_ A <-> A (_ A))
4	3	elrab 1422	. . 3 |- (A e. {x e. B | x (_ A} <-> (A e. B /\ A (_ A))
5	2, 4	sylibr 175	. 2 |- (A e. B -> A e. {x e. B | x (_ A})
6		sseq1 1521	. . . . . 6 |- (x = y -> (x (_ A <-> y (_ A))
7	6	elrab 1422	. . . . 5 |- (y e. {x e. B | x (_ A} <-> (y e. B /\ y (_ A))
8	7	pm3.27bd 263	. . . 4 |- (y e. {x e. B | x (_ A} -> y (_ A)
9	8	rgen 1247	. . 3 |- A.y e. {x e. B | x (_ A}y (_ A
10		ssunieq 1945	. . 3 |- ((A e. {x e. B | x (_ A} /\ A.y e. {x e. B | x (_ A}y (_ A) -> A = U.{x e. B | x (_ A})
11	9, 10	mpan2 519	. 2 |- (A e. {x e. B | x (_ A} -> A = U.{x e. B | x (_ A})
12	5, 11	syl 12	1 |- (A e. B -> A = U.{x e. B | x (_ A})

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201  {crab 1204   (_ wss 1487  U.cuni 1919

This theorem is referenced by:  chsupid 5312

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rab 1208  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920

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