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Theorem suplub 7971
 Description: A supremum is the least upper bound. See also supcl 7969 and supub 7970. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)
Hypotheses
Ref Expression
supmo.1
supcl.2
Assertion
Ref Expression
suplub
Distinct variable groups:   ,,,   ,,,   ,,,   ,
Allowed substitution hints:   (,,)   (,)

Proof of Theorem suplub
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpr 462 . . . . . . 7
2 breq1 4420 . . . . . . . . 9
3 breq1 4420 . . . . . . . . . 10
43rexbidv 2937 . . . . . . . . 9
52, 4imbi12d 321 . . . . . . . 8
65cbvralv 3053 . . . . . . 7
71, 6sylib 199 . . . . . 6
87a1i 11 . . . . 5
98ss2rabi 3540 . . . 4
10 supmo.1 . . . . . 6
1110supval2 7966 . . . . 5
12 supcl.2 . . . . . . 7
1310, 12supeu 7965 . . . . . 6
14 riotacl2 6271 . . . . . 6
1513, 14syl 17 . . . . 5
1611, 15eqeltrd 2508 . . . 4
179, 16sseldi 3459 . . 3
18 breq2 4421 . . . . . . 7
1918imbi1d 318 . . . . . 6
2019ralbidv 2862 . . . . 5
2120elrab 3226 . . . 4
2221simprbi 465 . . 3
2317, 22syl 17 . 2
24 breq1 4420 . . . . 5
25 breq1 4420 . . . . . 6
2625rexbidv 2937 . . . . 5
2724, 26imbi12d 321 . . . 4
2827rspccv 3176 . . 3
2928impd 432 . 2
3023, 29syl 17 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 370   wceq 1437   wcel 1867  wral 2773  wrex 2774  wreu 2775  crab 2777   class class class wbr 4417   wor 4765  crio 6257  csup 7951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-po 4766  df-so 4767  df-iota 5556  df-riota 6258  df-sup 7953 This theorem is referenced by:  suplub2  7972  supnub  7973  supiso  7988  infglb  8003  supxrun  11590  supxrunb1  11594  supxrunb2  11595  esum2d  28750  omssubaddlem  28957  omssubadd  28958  gtinf  30757
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