Theorem ssintub 4495

Description: Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem ssintub

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 4493	. 2 |- ( A C_ |^| { x e. B | A C_ x } <-> A. y e. { x e. B | A C_ x } A C_ y )
2		sseq2 3627	. . . 4 |- ( x = y -> ( A C_ x <-> A C_ y ) )
3	2	elrab 3363	. . 3 |- ( y e. { x e. B | A C_ x } <-> ( y e. B /\ A C_ y ) )
4	3	simprbi 480	. 2 |- ( y e. { x e. B | A C_ x } -> A C_ y )
5	1, 4	mprgbir 2927	1 |- A C_ |^| { x e. B | A C_ x }

Syntax hints:   e. wcel 1990   {crab 2916   C_ wss 3574   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-int 4476

This theorem is referenced by:  intmin  4497  wuncid  9565  mrcssid  16277  lspssid  18985  lbsextlem3  19160  aspssid  19333  sscls  20860  filufint  21724  spanss2  28204  shsval2i  28246  ococin  28267  chsupsn  28272  sssigagen  30208  dynkin  30230  igenss  33861  pclssidN  35181  dochocss  36655  rgspnssid  37740