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Theorem suplub 7940
 Description: A supremum is the least upper bound. See also supcl 7938 and supub 7939. (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:   ,,,   ,,,   ,,,   ,

Proof of Theorem suplub
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . . 7
2 breq1 4455 . . . . . . . . 9
3 breq1 4455 . . . . . . . . . 10
43rexbidv 2968 . . . . . . . . 9
52, 4imbi12d 320 . . . . . . . 8
65cbvralv 3084 . . . . . . 7
71, 6sylib 196 . . . . . 6
87a1i 11 . . . . 5
98ss2rabi 3581 . . . 4
10 supmo.1 . . . . . 6
1110supval2 7935 . . . . 5
12 supcl.2 . . . . . . 7
1310, 12supeu 7934 . . . . . 6
14 riotacl2 6271 . . . . . 6
1513, 14syl 16 . . . . 5
1611, 15eqeltrd 2545 . . . 4
179, 16sseldi 3501 . . 3
18 breq2 4456 . . . . . . 7
1918imbi1d 317 . . . . . 6
2019ralbidv 2896 . . . . 5
2120elrab 3257 . . . 4
2221simprbi 464 . . 3
2317, 22syl 16 . 2
24 breq1 4455 . . . . 5
25 breq1 4455 . . . . . 6
2625rexbidv 2968 . . . . 5
2724, 26imbi12d 320 . . . 4
2827rspccv 3207 . . 3
2928impd 431 . 2
3023, 29syl 16 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  E!wreu 2809  {crab 2811   class class class wbr 4452  Orwor 4804  iota_crio 6256  supcsup 7920 This theorem is referenced by:  suplub2  7941  supnub  7942  supiso  7954  supxrun  11536  supxrunb1  11540  supxrunb2  11541  gtinf  30137 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-po 4805  df-so 4806  df-iota 5556  df-riota 6257  df-sup 7921
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