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Theorem suprubex 8884
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 3156 . 2 (𝜑𝐵 ∈ ℝ)
4 lttri3 8014 . . . 4 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
54adantl 277 . . 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 6990 . 2 (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
85, 6supubti 6991 . . 3 (𝜑 → (𝐵𝐴 → ¬ sup(𝐴, ℝ, < ) < 𝐵))
92, 8mpd 13 . 2 (𝜑 → ¬ sup(𝐴, ℝ, < ) < 𝐵)
103, 7, 9nltled 8055 1 (𝜑𝐵 ≤ sup(𝐴, ℝ, < ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wcel 2148  wral 2455  wrex 2456  wss 3129   class class class wbr 4000  supcsup 6974  cr 7788   < clt 7969  cle 7970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-pre-ltirr 7901  ax-pre-apti 7904
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4628  df-cnv 4630  df-iota 5173  df-riota 5824  df-sup 6976  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975
This theorem is referenced by:  suprzclex  9327
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