Theorem ssintub 3888

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 3886	. 2 |- ( A C_ |^| { x e. B | A C_ x } <-> A. y e. { x e. B | A C_ x } A C_ y )
2		sseq2 3203	. . . 4 |- ( x = y -> ( A C_ x <-> A C_ y ) )
3	2	elrab 2916	. . 3 |- ( y e. { x e. B | A C_ x } <-> ( y e. B /\ A C_ y ) )
4	3	simprbi 275	. 2 |- ( y e. { x e. B | A C_ x } -> A C_ y )
5	1, 4	mprgbir 2552	1 |- A C_ |^| { x e. B | A C_ x }

Syntax hints:   e. wcel 2164   {crab 2476   C_ wss 3153   |^|cint 3870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-v 2762  df-in 3159  df-ss 3166  df-int 3871

This theorem is referenced by:  intmin  3890  lspssid  13896  sscls  14288