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Theorem ssintub 3624
 Description: Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
Assertion
Ref Expression
ssintub A {x BAx}
Distinct variable groups:   x,A   x,B

Proof of Theorem ssintub
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssint 3622 . 2 (A {x BAx} ↔ y {x BAx}Ay)
2 sseq2 2961 . . . 4 (x = y → (AxAy))
32elrab 2692 . . 3 (y {x BAx} ↔ (y B Ay))
43simprbi 260 . 2 (y {x BAx} → Ay)
51, 4mprgbir 2373 1 A {x BAx}
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1390  {crab 2304   ⊆ wss 2911  ∩ cint 3606 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-int 3607 This theorem is referenced by:  intmin  3626
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