Theorem ssintub 3892

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 3890	. 2 |- ( A C_ |^| { x e. B | A C_ x } <-> A. y e. { x e. B | A C_ x } A C_ y )
2		sseq2 3207	. . . 4 |- ( x = y -> ( A C_ x <-> A C_ y ) )
3	2	elrab 2920	. . 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 2555	1 |- A C_ |^| { x e. B | A C_ x }

Syntax hints:   e. wcel 2167   {crab 2479   C_ wss 3157   |^|cint 3874

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484  df-v 2765  df-in 3163  df-ss 3170  df-int 3875

This theorem is referenced by:  intmin  3894  lspssid  13956  sscls  14356