MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supub Structured version   Visualization version   GIF version

Theorem supub 9099
Description: A supremum is an upper bound. See also supcl 9098 and suplub 9100.

This proof demonstrates how to expand an iota-based definition (df-iota 6355) using riotacl2 7205.

(Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Hypotheses
Ref Expression
supmo.1 (𝜑𝑅 Or 𝐴)
supcl.2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
Assertion
Ref Expression
supub (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem supub
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simpl 486 . . . . . 6 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)
21a1i 11 . . . . 5 (𝑥𝐴 → ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐵 ¬ 𝑥𝑅𝑦))
32ss2rabi 4004 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ⊆ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦}
4 supmo.1 . . . . . 6 (𝜑𝑅 Or 𝐴)
54supval2 9095 . . . . 5 (𝜑 → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
6 supcl.2 . . . . . . 7 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
74, 6supeu 9094 . . . . . 6 (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
8 riotacl2 7205 . . . . . 6 (∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
97, 8syl 17 . . . . 5 (𝜑 → (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
105, 9eqeltrd 2839 . . . 4 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
113, 10sselid 3912 . . 3 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦})
12 breq2 5071 . . . . . . . 8 (𝑦 = 𝑤 → (𝑥𝑅𝑦𝑥𝑅𝑤))
1312notbid 321 . . . . . . 7 (𝑦 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑤))
1413cbvralvw 3370 . . . . . 6 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤𝐵 ¬ 𝑥𝑅𝑤)
15 breq1 5070 . . . . . . . 8 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑥𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1615notbid 321 . . . . . . 7 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (¬ 𝑥𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1716ralbidv 3119 . . . . . 6 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤𝐵 ¬ 𝑥𝑅𝑤 ↔ ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1814, 17syl5bb 286 . . . . 5 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1918elrab 3614 . . . 4 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2019simprbi 500 . . 3 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦} → ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
2111, 20syl 17 . 2 (𝜑 → ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
22 breq2 5071 . . . 4 (𝑤 = 𝐶 → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
2322notbid 321 . . 3 (𝑤 = 𝐶 → (¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
2423rspccv 3546 . 2 (∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
2521, 24syl 17 1 (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wcel 2111  wral 3062  wrex 3063  ∃!wreu 3064  {crab 3066   class class class wbr 5067   Or wor 5481  crio 7187  supcsup 9080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3422  df-sbc 3709  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-po 5482  df-so 5483  df-iota 6355  df-riota 7188  df-sup 9082
This theorem is referenced by:  suplub2  9101  supgtoreq  9110  supiso  9115  inflb  9129  suprub  11817  suprzub  12559  supxrun  12930  supxrub  12938  dgrub  25152  supssd  30788  ssnnssfz  30852  oddpwdc  32057  itg2addnclem  35591  supubt  35660  ssnn0ssfz  45386
  Copyright terms: Public domain W3C validator