Theorem supssd 31044

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 9217	. 2 ⊢ (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
4		supssd.1	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
5	4	sseld 3920	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐶))
6	1, 2	supub 9218	. . . 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 3102	. 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 9221	. 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 696	1 ⊢ (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887   class class class wbr 5074   Or wor 5502  supcsup 9199

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-po 5503  df-so 5504  df-iota 6391  df-riota 7232  df-sup 9201

This theorem is referenced by:  xrsupssd  31082