Theorem supssd 30764

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 9074	. 2 ⊢ (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
4		supssd.1	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
5	4	sseld 3900	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐶))
6	1, 2	supub 9075	. . . 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 3104	. 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 9078	. 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 699	1 ⊢ (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062   ⊆ wss 3866   class class class wbr 5053   Or wor 5467  supcsup 9056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-po 5468  df-so 5469  df-iota 6338  df-riota 7170  df-sup 9058

This theorem is referenced by:  xrsupssd  30802