Theorem ssintub 3862

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 3860	. 2 |- ( A C_ |^| { x e. B | A C_ x } <-> A. y e. { x e. B | A C_ x } A C_ y )
2		sseq2 3179	. . . 4 |- ( x = y -> ( A C_ x <-> A C_ y ) )
3	2	elrab 2893	. . 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 2535	1 |- A C_ |^| { x e. B | A C_ x }

Syntax hints:   e. wcel 2148   {crab 2459   C_ wss 3129   |^|cint 3844

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2739  df-in 3135  df-ss 3142  df-int 3845

This theorem is referenced by:  intmin  3864  sscls  13482