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Mathbox for Thierry Arnoux |
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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 8716 | . 2 ⊢ (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴) |
4 | supssd.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
5 | 4 | sseld 3852 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐶)) |
6 | 1, 2 | supub 8717 | . . . 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 3126 | . 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 8720 | . 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 687 | 1 ⊢ (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∈ wcel 2051 ∀wral 3083 ∃wrex 3084 ⊆ wss 3824 class class class wbr 4926 Or wor 5322 supcsup 8698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-po 5323 df-so 5324 df-iota 6150 df-riota 6936 df-sup 8700 |
This theorem is referenced by: xrsupssd 30260 |
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