Theorem supssd 9378

Description: Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Hypotheses

Ref	Expression
supssd.0	⊢ (𝜑 → 𝑅 Or 𝐴)
supssd.1	⊢ (𝜑 → 𝐵 ⊆ 𝐶)
supssd.2	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
supssd.3	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
supssd.4	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 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

Step	Hyp	Ref	Expression
1		supssd.0	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
2		supssd.4	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))
3	1, 2	supcl 9373	. 2 ⊢ (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
4		supssd.1	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
5	4	sseld 3934	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐶))
6	1, 2	supub 9374	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧))
7	5, 6	syld 47	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐵 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧))
8	7	ralrimiv 3129	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧)
9		supssd.3	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
10	1, 9	supnub 9377	. 2 ⊢ (𝜑 → ((sup(𝐶, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑧) → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅)))
11	3, 8, 10	mp2and 700	1 ⊢ (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903   class class class wbr 5100   Or wor 5539  supcsup 9355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-po 5540  df-so 5541  df-iota 6456  df-riota 7325  df-sup 9357

This theorem is referenced by:  xrsupssd  13260