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Theorem seinf 25251
 Description: The least element of a poset is the infimum of the poset. (Contributed by FL, 19-Sep-2011.)
Hypothesis
Ref Expression
seinf.1
Assertion
Ref Expression
seinf leR

Proof of Theorem seinf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2
2 id 19 . . . 4
3 dmeq 4879 . . . . 5
4 seinf.1 . . . . 5
53, 4syl6eqr 2333 . . . 4
62, 5oveq12d 5876 . . 3
7 df-ler 25249 . . 3 leR
8 ovex 5883 . . 3
96, 7, 8fvmpt 5602 . 2 leR
101, 9syl 15 1 leR
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1623   wcel 1684  cvv 2788   cdm 4689  cfv 5255  (class class class)co 5858   cinf 14304  leRcse 25221 This theorem is referenced by:  sege  25252 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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 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-ral 2548  df-rex 2549  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-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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-ler 25249
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