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Theorem suprubex 8305
Description: A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)
Hypotheses
Ref Expression
suprubex.ex (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
suprubex.ss (𝜑𝐴 ⊆ ℝ)
suprubex.b (𝜑𝐵𝐴)
Assertion
Ref Expression
suprubex (𝜑𝐵 ≤ sup(𝐴, ℝ, < ))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)

Proof of Theorem suprubex
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suprubex.ss . . 3 (𝜑𝐴 ⊆ ℝ)
2 suprubex.b . . 3 (𝜑𝐵𝐴)
31, 2sseldd 3011 . 2 (𝜑𝐵 ∈ ℝ)
4 lttri3 7467 . . . 4 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
54adantl 271 . . 3 ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
6 suprubex.ex . . 3 (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
75, 6supclti 6599 . 2 (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
85, 6supubti 6600 . . 3 (𝜑 → (𝐵𝐴 → ¬ sup(𝐴, ℝ, < ) < 𝐵))
92, 8mpd 13 . 2 (𝜑 → ¬ sup(𝐴, ℝ, < ) < 𝐵)
103, 7, 9nltled 7506 1 (𝜑𝐵 ≤ sup(𝐴, ℝ, < ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wcel 1434  wral 2353  wrex 2354  wss 2984   class class class wbr 3811  supcsup 6583  cr 7251   < clt 7424  cle 7425
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-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-cnex 7338  ax-resscn 7339  ax-pre-ltirr 7359  ax-pre-apti 7362
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-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-xp 4406  df-cnv 4408  df-iota 4933  df-riota 5546  df-sup 6585  df-pnf 7426  df-mnf 7427  df-xr 7428  df-ltxr 7429  df-le 7430
This theorem is referenced by:  suprzclex  8739
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