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Theorem supssd 29796
 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 8529 . 2 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
4 supssd.1 . . . . 5 (𝜑𝐵𝐶)
54sseld 3743 . . . 4 (𝜑 → (𝑧𝐵𝑧𝐶))
61, 2supub 8530 . . . 4 (𝜑 → (𝑧𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧))
75, 6syld 47 . . 3 (𝜑 → (𝑧𝐵 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧))
87ralrimiv 3103 . 2 (𝜑 → ∀𝑧𝐵 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧)
9 supssd.3 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
101, 9supnub 8533 . 2 (𝜑 → ((sup(𝐶, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑧𝐵 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧) → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅)))
113, 8, 10mp2and 717 1 (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ⊆ wss 3715   class class class wbr 4804   Or wor 5186  supcsup 8511 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-po 5187  df-so 5188  df-iota 6012  df-riota 6774  df-sup 8513 This theorem is referenced by:  xrsupssd  29833
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