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Theorem ssintub 3944
 Description: Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
Assertion
Ref Expression
ssintub A {x B A x}
Distinct variable groups:   x,A   x,B

Proof of Theorem ssintub
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssint 3942 . 2 (A {x B A x} ↔ y {x B A x}A y)
2 sseq2 3293 . . . 4 (x = y → (A xA y))
32elrab 2994 . . 3 (y {x B A x} ↔ (y B A y))
43simprbi 450 . 2 (y {x B A x} → A y)
51, 4mprgbir 2684 1 A {x B A x}
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1710  {crab 2618   ⊆ wss 3257  ∩cint 3926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927 This theorem is referenced by:  intmin  3946
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