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Mirrors > Home > MPE Home > Th. List > Mathboxes > infssd | Structured version Visualization version GIF version |
Description: Inequality deduction for infimum of a subset. (Contributed by AV, 4-Oct-2020.) |
Ref | Expression |
---|---|
infssd.0 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
infssd.1 | ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
infssd.3 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦))) |
infssd.4 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
Ref | Expression |
---|---|
infssd | ⊢ (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infssd.0 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
2 | infssd.4 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | |
3 | 1, 2 | infcl 8277 | . 2 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
4 | infssd.1 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
5 | 4 | sseld 3567 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵)) |
6 | 1, 2 | inflb 8278 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐵 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅))) |
7 | 5, 6 | syld 46 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐶 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅))) |
8 | 7 | ralrimiv 2948 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐶 ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)) |
9 | infssd.3 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦))) | |
10 | 1, 9 | infnlb 8281 | . 2 ⊢ (𝜑 → ((inf(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐶 ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)) → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅))) |
11 | 3, 8, 10 | mp2and 711 | 1 ⊢ (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 class class class wbr 4583 Or wor 4958 infcinf 8230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-po 4959 df-so 4960 df-cnv 5046 df-iota 5768 df-riota 6511 df-sup 8231 df-inf 8232 |
This theorem is referenced by: xrge0infssd 28916 |
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