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Theorem suprleubex 8841
Description: The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
Hypotheses
Ref Expression
suprubex.ex (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
suprubex.ss (𝜑𝐴 ⊆ ℝ)
suprlubex.b (𝜑𝐵 ∈ ℝ)
Assertion
Ref Expression
suprleubex (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 𝑧𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥   𝑧,𝐵
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑥,𝑦)

Proof of Theorem suprleubex
Dummy variables 𝑓 𝑔 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lttri3 7970 . . . . . . . 8 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
21adantl 275 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
3 suprubex.ex . . . . . . 7 (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
42, 3supclti 6955 . . . . . 6 (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
5 suprlubex.b . . . . . 6 (𝜑𝐵 ∈ ℝ)
64, 5lenltd 8008 . . . . 5 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup(𝐴, ℝ, < )))
7 suprubex.ss . . . . . 6 (𝜑𝐴 ⊆ ℝ)
83, 7, 5suprnubex 8840 . . . . 5 (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))
96, 8bitrd 187 . . . 4 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))
10 breq2 3981 . . . . . 6 (𝑤 = 𝑧 → (𝐵 < 𝑤𝐵 < 𝑧))
1110notbid 657 . . . . 5 (𝑤 = 𝑧 → (¬ 𝐵 < 𝑤 ↔ ¬ 𝐵 < 𝑧))
1211cbvralv 2690 . . . 4 (∀𝑤𝐴 ¬ 𝐵 < 𝑤 ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧)
139, 12bitr4di 197 . . 3 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤𝐴 ¬ 𝐵 < 𝑤))
147sselda 3138 . . . . 5 ((𝜑𝑤𝐴) → 𝑤 ∈ ℝ)
155adantr 274 . . . . 5 ((𝜑𝑤𝐴) → 𝐵 ∈ ℝ)
1614, 15lenltd 8008 . . . 4 ((𝜑𝑤𝐴) → (𝑤𝐵 ↔ ¬ 𝐵 < 𝑤))
1716ralbidva 2460 . . 3 (𝜑 → (∀𝑤𝐴 𝑤𝐵 ↔ ∀𝑤𝐴 ¬ 𝐵 < 𝑤))
1813, 17bitr4d 190 . 2 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤𝐴 𝑤𝐵))
19 breq1 3980 . . 3 (𝑤 = 𝑧 → (𝑤𝐵𝑧𝐵))
2019cbvralv 2690 . 2 (∀𝑤𝐴 𝑤𝐵 ↔ ∀𝑧𝐴 𝑧𝐵)
2118, 20bitrdi 195 1 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 𝑧𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wcel 2135  wral 2442  wrex 2443  wss 3112   class class class wbr 3977  supcsup 6939  cr 7744   < clt 7925  cle 7926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-cnex 7836  ax-resscn 7837  ax-pre-ltirr 7857  ax-pre-ltwlin 7858  ax-pre-lttrn 7859  ax-pre-apti 7860
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2724  df-sbc 2948  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-po 4269  df-iso 4270  df-xp 4605  df-cnv 4607  df-iota 5148  df-riota 5793  df-sup 6941  df-pnf 7927  df-mnf 7928  df-xr 7929  df-ltxr 7930  df-le 7931
This theorem is referenced by:  suprzclex  9281  suplociccex  13170
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