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Theorem lbinfle 8409
Description: If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
Assertion
Ref Expression
lbinfle ((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦𝐴𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴)
Distinct variable groups:   𝑥,𝑆,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem lbinfle
StepHypRef Expression
1 lbinf 8407 . . 3 ((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → inf(𝑆, ℝ, < ) = (𝑥𝑆𝑦𝑆 𝑥𝑦))
213adant3 963 . 2 ((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦𝐴𝑆) → inf(𝑆, ℝ, < ) = (𝑥𝑆𝑦𝑆 𝑥𝑦))
3 lble 8406 . 2 ((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦𝐴𝑆) → (𝑥𝑆𝑦𝑆 𝑥𝑦) ≤ 𝐴)
42, 3eqbrtrd 3865 1 ((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦𝐴𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 924   = wceq 1289  wcel 1438  wral 2359  wrex 2360  wss 2999   class class class wbr 3845  crio 5607  infcinf 6676  cr 7347   < clt 7520  cle 7521
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-pre-ltirr 7455  ax-pre-apti 7458
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-iota 4980  df-riota 5608  df-sup 6677  df-inf 6678  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526
This theorem is referenced by: (None)
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