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Theorem supssd 31680
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 9402 . 2 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
4 supssd.1 . . . . 5 (𝜑𝐵𝐶)
54sseld 3947 . . . 4 (𝜑 → (𝑧𝐵𝑧𝐶))
61, 2supub 9403 . . . 4 (𝜑 → (𝑧𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧))
75, 6syld 47 . . 3 (𝜑 → (𝑧𝐵 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧))
87ralrimiv 3139 . 2 (𝜑 → ∀𝑧𝐵 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧)
9 supssd.3 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
101, 9supnub 9406 . 2 (𝜑 → ((sup(𝐶, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑧𝐵 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧) → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅)))
113, 8, 10mp2and 698 1 (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wcel 2107  wral 3061  wrex 3070  wss 3914   class class class wbr 5109   Or wor 5548  supcsup 9384
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-po 5549  df-so 5550  df-iota 6452  df-riota 7317  df-sup 9386
This theorem is referenced by:  xrsupssd  31718
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