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Theorem infminti 6535
 Description: The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.)
Hypotheses
Ref Expression
infminti.ti
infminti.2
infminti.3
infminti.4
Assertion
Ref Expression
infminti inf
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,

Proof of Theorem infminti
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 infminti.ti . 2
2 infminti.2 . 2
3 infminti.4 . 2
4 infminti.3 . . 3
5 simprr 499 . . 3
6 breq1 3809 . . . 4
76rspcev 2710 . . 3
84, 5, 7syl2an2r 560 . 2
91, 2, 3, 8eqinftid 6529 1 inf
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 102   wb 103   wceq 1285   wcel 1434  wrex 2354   class class class wbr 3806  infcinf 6491 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-cnv 4400  df-iota 4918  df-riota 5520  df-sup 6492  df-inf 6493 This theorem is referenced by:  lbinf  8129  lcmgcdlem  10650
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