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Theorem suprleubex 8724
 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 7856 . . . . . . . 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 6885 . . . . . 6 (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
5 suprlubex.b . . . . . 6 (𝜑𝐵 ∈ ℝ)
64, 5lenltd 7892 . . . . 5 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup(𝐴, ℝ, < )))
7 suprubex.ss . . . . . 6 (𝜑𝐴 ⊆ ℝ)
83, 7, 5suprnubex 8723 . . . . 5 (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))
96, 8bitrd 187 . . . 4 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))
10 breq2 3933 . . . . . 6 (𝑤 = 𝑧 → (𝐵 < 𝑤𝐵 < 𝑧))
1110notbid 656 . . . . 5 (𝑤 = 𝑧 → (¬ 𝐵 < 𝑤 ↔ ¬ 𝐵 < 𝑧))
1211cbvralv 2654 . . . 4 (∀𝑤𝐴 ¬ 𝐵 < 𝑤 ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧)
139, 12syl6bbr 197 . . 3 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤𝐴 ¬ 𝐵 < 𝑤))
147sselda 3097 . . . . 5 ((𝜑𝑤𝐴) → 𝑤 ∈ ℝ)
155adantr 274 . . . . 5 ((𝜑𝑤𝐴) → 𝐵 ∈ ℝ)
1614, 15lenltd 7892 . . . 4 ((𝜑𝑤𝐴) → (𝑤𝐵 ↔ ¬ 𝐵 < 𝑤))
1716ralbidva 2433 . . 3 (𝜑 → (∀𝑤𝐴 𝑤𝐵 ↔ ∀𝑤𝐴 ¬ 𝐵 < 𝑤))
1813, 17bitr4d 190 . 2 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤𝐴 𝑤𝐵))
19 breq1 3932 . . 3 (𝑤 = 𝑧 → (𝑤𝐵𝑧𝐵))
2019cbvralv 2654 . 2 (∀𝑤𝐴 𝑤𝐵 ↔ ∀𝑧𝐴 𝑧𝐵)
2118, 20syl6bb 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 1480  ∀wral 2416  ∃wrex 2417   ⊆ wss 3071   class class class wbr 3929  supcsup 6869  ℝcr 7631   < clt 7812   ≤ cle 7813 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-po 4218  df-iso 4219  df-xp 4545  df-cnv 4547  df-iota 5088  df-riota 5730  df-sup 6871  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818 This theorem is referenced by:  suprzclex  9161  suplociccex  12786
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