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Theorem suprleubex 9026
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 8151 . . . . . . . 8 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
21adantl 277 . . . . . . 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 7099 . . . . . 6 (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
5 suprlubex.b . . . . . 6 (𝜑𝐵 ∈ ℝ)
64, 5lenltd 8189 . . . . 5 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup(𝐴, ℝ, < )))
7 suprubex.ss . . . . . 6 (𝜑𝐴 ⊆ ℝ)
83, 7, 5suprnubex 9025 . . . . 5 (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))
96, 8bitrd 188 . . . 4 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))
10 breq2 4047 . . . . . 6 (𝑤 = 𝑧 → (𝐵 < 𝑤𝐵 < 𝑧))
1110notbid 668 . . . . 5 (𝑤 = 𝑧 → (¬ 𝐵 < 𝑤 ↔ ¬ 𝐵 < 𝑧))
1211cbvralv 2737 . . . 4 (∀𝑤𝐴 ¬ 𝐵 < 𝑤 ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧)
139, 12bitr4di 198 . . 3 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤𝐴 ¬ 𝐵 < 𝑤))
147sselda 3192 . . . . 5 ((𝜑𝑤𝐴) → 𝑤 ∈ ℝ)
155adantr 276 . . . . 5 ((𝜑𝑤𝐴) → 𝐵 ∈ ℝ)
1614, 15lenltd 8189 . . . 4 ((𝜑𝑤𝐴) → (𝑤𝐵 ↔ ¬ 𝐵 < 𝑤))
1716ralbidva 2501 . . 3 (𝜑 → (∀𝑤𝐴 𝑤𝐵 ↔ ∀𝑤𝐴 ¬ 𝐵 < 𝑤))
1813, 17bitr4d 191 . 2 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤𝐴 𝑤𝐵))
19 breq1 4046 . . 3 (𝑤 = 𝑧 → (𝑤𝐵𝑧𝐵))
2019cbvralv 2737 . 2 (∀𝑤𝐴 𝑤𝐵 ↔ ∀𝑧𝐴 𝑧𝐵)
2118, 20bitrdi 196 1 (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 𝑧𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wcel 2175  wral 2483  wrex 2484  wss 3165   class class class wbr 4043  supcsup 7083  cr 7923   < clt 8106  cle 8107
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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-po 4342  df-iso 4343  df-xp 4680  df-cnv 4682  df-iota 5231  df-riota 5898  df-sup 7085  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112
This theorem is referenced by:  suprzclex  9470  suplociccex  15068
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