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Theorem supval2 7935
Description: Alternative expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Thierry Arnoux, 24-Sep-2017.)
Hypothesis
Ref Expression
supmo.1
Assertion
Ref Expression
supval2
Distinct variable groups:   , , ,   , , ,   , , ,

Proof of Theorem supval2
StepHypRef Expression
1 supmo.1 . 2
2 simpl 457 . . . . . 6
3 simpr 461 . . . . . 6
42, 3supeu 7934 . . . . 5
5 riotauni 6263 . . . . 5
64, 5syl 16 . . . 4
7 df-sup 7921 . . . 4
86, 7syl6reqr 2517 . . 3
9 rabn0 3805 . . . . . . . . . 10
109necon1bbii 2721 . . . . . . . . 9
1110biimpi 194 . . . . . . . 8
1211unieqd 4259 . . . . . . 7
13 uni0 4276 . . . . . . 7
1412, 13syl6eq 2514 . . . . . 6
157, 14syl5eq 2510 . . . . 5
16 reurex 3074 . . . . . . 7
1716con3i 135 . . . . . 6
18 riotaund 6293 . . . . . 6
1917, 18syl 16 . . . . 5
2015, 19eqtr4d 2501 . . . 4
2120adantl 466 . . 3
228, 21pm2.61dan 791 . 2
231, 22syl 16 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  A.wral 2807  E.wrex 2808  E!wreu 2809  {crab 2811   c0 3784  U.cuni 4249   class class class wbr 4452  Orwor 4804  iota_crio 6256  supcsup 7920
This theorem is referenced by:  eqsup  7936  supcl  7938  supub  7939  suplub  7940  fisupcl  7948  toslub  27656  tosglb  27658
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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