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Theorem supssd 32429
Description: Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
Hypotheses
Ref Expression
supssd.0 (𝜑𝑅 Or 𝐴)
supssd.1 (𝜑𝐵𝐶)
supssd.2 (𝜑𝐶𝐴)
supssd.3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
supssd.4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
Assertion
Ref Expression
supssd (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem supssd
StepHypRef Expression
1 supssd.0 . . 3 (𝜑𝑅 Or 𝐴)
2 supssd.4 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
31, 2supcl 9450 . 2 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
4 supssd.1 . . . . 5 (𝜑𝐵𝐶)
54sseld 3974 . . . 4 (𝜑 → (𝑧𝐵𝑧𝐶))
61, 2supub 9451 . . . 4 (𝜑 → (𝑧𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧))
75, 6syld 47 . . 3 (𝜑 → (𝑧𝐵 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧))
87ralrimiv 3137 . 2 (𝜑 → ∀𝑧𝐵 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧)
9 supssd.3 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
101, 9supnub 9454 . 2 (𝜑 → ((sup(𝐶, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑧𝐵 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧) → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅)))
113, 8, 10mp2and 696 1 (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2098  wral 3053  wrex 3062  wss 3941   class class class wbr 5139   Or wor 5578  supcsup 9432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-po 5579  df-so 5580  df-iota 6486  df-riota 7358  df-sup 9434
This theorem is referenced by:  xrsupssd  32467
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