Theorem ssintub 2555

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 2553	. 2 |- (A (_ |^|{x e. B | A (_ x} <-> A.y e. {x e. B | A (_ x}A (_ y)
2		sseq2 2086	. . . 4 |- (x = y -> (A (_ x <-> A (_ y))
3	2	elrab 1908	. . 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 1704	1 |- A (_ |^|{x e. B | A (_ x}

Syntax hints:   e. wcel 960  {crab 1651   (_ wss 2050  |^|cint 2537

This theorem is referenced by:  intmin 2557  sscls 7686  ococint 9292  chsupsn 9307  hsupunss 9308  spanss2 9309  shsumval2 9355

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-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rab 1655  df-v 1815  df-in 2054  df-ss 2056  df-int 2538

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