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Theorem suprleubex 8512
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 7662 . . . . . . . 8 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
21adantl 272 . . . . . . 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 6773 . . . . . 6 (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
5 suprlubex.b . . . . . 6 (𝜑𝐵 ∈ ℝ)
64, 5lenltd 7698 . . . . 5 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup(𝐴, ℝ, < )))
7 suprubex.ss . . . . . 6 (𝜑𝐴 ⊆ ℝ)
83, 7, 5suprnubex 8511 . . . . 5 (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))
96, 8bitrd 187 . . . 4 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))
10 breq2 3871 . . . . . 6 (𝑤 = 𝑧 → (𝐵 < 𝑤𝐵 < 𝑧))
1110notbid 630 . . . . 5 (𝑤 = 𝑧 → (¬ 𝐵 < 𝑤 ↔ ¬ 𝐵 < 𝑧))
1211cbvralv 2604 . . . 4 (∀𝑤𝐴 ¬ 𝐵 < 𝑤 ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧)
139, 12syl6bbr 197 . . 3 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤𝐴 ¬ 𝐵 < 𝑤))
147sselda 3039 . . . . 5 ((𝜑𝑤𝐴) → 𝑤 ∈ ℝ)
155adantr 271 . . . . 5 ((𝜑𝑤𝐴) → 𝐵 ∈ ℝ)
1614, 15lenltd 7698 . . . 4 ((𝜑𝑤𝐴) → (𝑤𝐵 ↔ ¬ 𝐵 < 𝑤))
1716ralbidva 2387 . . 3 (𝜑 → (∀𝑤𝐴 𝑤𝐵 ↔ ∀𝑤𝐴 ¬ 𝐵 < 𝑤))
1813, 17bitr4d 190 . 2 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤𝐴 𝑤𝐵))
19 breq1 3870 . . 3 (𝑤 = 𝑧 → (𝑤𝐵𝑧𝐵))
2019cbvralv 2604 . 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 1445  wral 2370  wrex 2371  wss 3013   class class class wbr 3867  supcsup 6757  cr 7446   < clt 7619  cle 7620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-po 4147  df-iso 4148  df-xp 4473  df-cnv 4475  df-iota 5014  df-riota 5646  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625
This theorem is referenced by:  suprzclex  8943
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