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Mirrors > Home > MPE Home > Th. List > Mathboxes > supssd | Structured version Visualization version GIF version |
Description: Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
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))) |
Ref | Expression |
---|---|
supssd | ⊢ (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅)) |
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 8916 | . 2 ⊢ (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴) |
4 | supssd.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
5 | 4 | sseld 3966 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐶)) |
6 | 1, 2 | supub 8917 | . . . 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 3181 | . 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 8920 | . 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 697 | 1 ⊢ (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 class class class wbr 5059 Or wor 5468 supcsup 8898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-po 5469 df-so 5470 df-iota 6309 df-riota 7108 df-sup 8900 |
This theorem is referenced by: xrsupssd 30477 |
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