Theorem ssintub 1981

Description: Subclass of a least upper bound.

Assertion

Ref	Expression
ssintub	|- A (_ |^|{x e. B | A (_ x}

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

Proof of Theorem ssintub

Step	Hyp	Ref	Expression
1		ssint 1980	. 2 |- (A (_ |^|{x e. B | A (_ x} <-> A.y e. {x e. B | A (_ x}A (_ y)
2		sseq2 1522	. . . 4 |- (x = y -> (A (_ x <-> A (_ y))
3	2	elrab 1422	. . 3 |- (y e. {x e. B | A (_ x} <-> (y e. B /\ A (_ y))
4	3	pm3.27bd 263	. 2 |- (y e. {x e. B | A (_ x} -> A (_ y)
5	1, 4	mprgbir 1250	1 |- A (_ |^|{x e. B | A (_ x}

Syntax hints:   e. wcel 1092  {crab 1204   (_ wss 1487  |^|cint 1965

This theorem is referenced by:  intmin 1982  ococint 5298  chsupsn 5313  hsupunss 5314  spanss2 5315  shsumval2 5361

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-int 1966

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