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Theorem supdef 25262
 Description: If it exists, a supremum of is greater or equal to every element of and is the least upper bound of . Here the existence of the supremum is expressed by the idiom . (Contributed by FL, 23-May-2011.)
Hypothesis
Ref Expression
supdef.1
Assertion
Ref Expression
supdef
Distinct variable groups:   ,,   ,,   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem supdef
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 supdef.1 . . . . 5
21spwval 14334 . . . 4
4 dmexg 4939 . . . . . . 7
51, 4syl5eqel 2367 . . . . . 6
653ad2ant1 976 . . . . 5
7 simp3 957 . . . . . 6
83, 7eqeltrrd 2358 . . . . 5
9 riotaclbg 6344 . . . . . 6
109biimpar 471 . . . . 5
116, 8, 10syl2anc 642 . . . 4
12 riotacl2 6318 . . . 4
1311, 12syl 15 . . 3
143, 13eqeltrd 2357 . 2
15 breq2 4027 . . . . . 6
1615ralbidv 2563 . . . . 5
17 breq1 4026 . . . . . . 7
1817imbi2d 307 . . . . . 6
1918ralbidv 2563 . . . . 5
2016, 19anbi12d 691 . . . 4
2120elrab 2923 . . 3
2221simprbi 450 . 2
2314, 22syl 15 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1623   wcel 1684  wral 2543  wreu 2545  crab 2547  cvv 2788   class class class wbr 4023   cdm 4689  (class class class)co 5858  crio 6297  cps 14301   cspw 14303 This theorem is referenced by:  supdefa  25263  defge3  25271  supaub  25273  supwlub  25274  supnuf  25629 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-undef 6298  df-riota 6304  df-ps 14306  df-spw 14308
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