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Theorem supub 9454
Description: A supremum is an upper bound. See also supcl 9453 and suplub 9455.

This proof demonstrates how to expand an iota-based definition (df-iota 6496) using riotacl2 7382.

(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 484 . . . . . 6 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)
21a1i 11 . . . . 5 (𝑥𝐴 → ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐵 ¬ 𝑥𝑅𝑦))
32ss2rabi 4075 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ⊆ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦}
4 supmo.1 . . . . . 6 (𝜑𝑅 Or 𝐴)
54supval2 9450 . . . . 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 9449 . . . . . 6 (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
8 riotacl2 7382 . . . . . 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 2834 . . . 4 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
113, 10sselid 3981 . . 3 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦})
12 breq2 5153 . . . . . . . 8 (𝑦 = 𝑤 → (𝑥𝑅𝑦𝑥𝑅𝑤))
1312notbid 318 . . . . . . 7 (𝑦 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑤))
1413cbvralvw 3235 . . . . . 6 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤𝐵 ¬ 𝑥𝑅𝑤)
15 breq1 5152 . . . . . . . 8 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑥𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1615notbid 318 . . . . . . 7 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (¬ 𝑥𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1716ralbidv 3178 . . . . . 6 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤𝐵 ¬ 𝑥𝑅𝑤 ↔ ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1814, 17bitrid 283 . . . . 5 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1918elrab 3684 . . . 4 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2019simprbi 498 . . 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 5153 . . . 4 (𝑤 = 𝐶 → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
2322notbid 318 . . 3 (𝑤 = 𝐶 → (¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
2423rspccv 3610 . 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 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  ∃!wreu 3375  {crab 3433   class class class wbr 5149   Or wor 5588  crio 7364  supcsup 9435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-po 5589  df-so 5590  df-iota 6496  df-riota 7365  df-sup 9437
This theorem is referenced by:  suplub2  9456  supgtoreq  9465  supiso  9470  inflb  9484  suprub  12175  suprzub  12923  supxrun  13295  supxrub  13303  dgrub  25748  supssd  31934  ssnnssfz  31998  oddpwdc  33353  itg2addnclem  36539  supubt  36607  ssnn0ssfz  47025
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