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Theorem suplub 7466
 Description: A supremum is the least upper bound. See also supcl 7464 and supub 7465. (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 449 . . . . . . 7
2 breq1 4216 . . . . . . . . 9
3 breq1 4216 . . . . . . . . . 10
43rexbidv 2727 . . . . . . . . 9
52, 4imbi12d 313 . . . . . . . 8
65cbvralv 2933 . . . . . . 7
71, 6sylib 190 . . . . . 6
87a1i 11 . . . . 5
98ss2rabi 3426 . . . 4
10 supmo.1 . . . . . 6
11 supcl.2 . . . . . 6
1210, 11supval2 7461 . . . . 5
1310, 11supeu 7460 . . . . . 6
14 riotacl2 6564 . . . . . 6
1513, 14syl 16 . . . . 5
1612, 15eqeltrd 2511 . . . 4
179, 16sseldi 3347 . . 3
18 breq2 4217 . . . . . . 7
1918imbi1d 310 . . . . . 6
2019ralbidv 2726 . . . . 5
2120elrab 3093 . . . 4
2221simprbi 452 . . 3
2317, 22syl 16 . 2
24 breq1 4216 . . . . 5
25 breq1 4216 . . . . . 6
2625rexbidv 2727 . . . . 5
2724, 26imbi12d 313 . . . 4
2827rspccv 3050 . . 3
2928imp3a 422 . 2
3023, 29syl 16 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   wceq 1653   wcel 1726  wral 2706  wrex 2707  wreu 2708  crab 2710   class class class wbr 4213   wor 4503  crio 6543  csup 7446 This theorem is referenced by:  suplub2  7467  supnub  7468  supiso  7478  supxrun  10895  supxrunb1  10899  supxrunb2  10900  gtinf  26323 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-po 4504  df-so 4505  df-iota 5419  df-riota 6550  df-sup 7447
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