Theorem ssintub 2551

Description: Subclass of a least upper bound.

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem ssintub

Step	Hyp	Ref	Expression
1		ssint 2549	. 2 |- (A (_ |^|{x e. B | A (_ x} <-> A.y e. {x e. B | A (_ x}A (_ y)
2		sseq2 2083	. . . 4 |- (x = y -> (A (_ x <-> A (_ y))
3	2	elrab 1905	. . 3 |- (y e. {x e. B | A (_ x} <-> (y e. B /\ A (_ y))
4	3	pm3.27bi 326	. 2 |- (y e. {x e. B | A (_ x} -> A (_ y)
5	1, 4	mprgbir 1701	1 |- A (_ |^|{x e. B | A (_ x}

Syntax hints:   e. wcel 958  {crab 1648   (_ wss 2047  |^|cint 2533

This theorem is referenced by:  intmin 2553  sscls 7689  ococint 9297  chsupsn 9312  hsupunss 9313  spanss2 9314  shsumval2 9360

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rab 1652  df-v 1812  df-in 2051  df-ss 2053  df-int 2534

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