Theorem ssintub 3917

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 3915	. 2 |- ( A C_ |^| { x e. B | A C_ x } <-> A. y e. { x e. B | A C_ x } A C_ y )
2		sseq2 3225	. . . 4 |- ( x = y -> ( A C_ x <-> A C_ y ) )
3	2	elrab 2936	. . 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 2566	1 |- A C_ |^| { x e. B | A C_ x }

Syntax hints:   e. wcel 2178   {crab 2490   C_ wss 3174   |^|cint 3899

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-v 2778  df-in 3180  df-ss 3187  df-int 3900

This theorem is referenced by:  intmin  3919  lspssid  14277  sscls  14707