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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 9480 | . 2 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
4 | infssd.1 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
5 | 4 | sseld 3974 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵)) |
6 | 1, 2 | inflb 9481 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐵 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅))) |
7 | 5, 6 | syld 47 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐶 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅))) |
8 | 7 | ralrimiv 3137 | . 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 9484 | . 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 696 | 1 ⊢ (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 ⊆ wss 3941 class class class wbr 5139 Or wor 5578 infcinf 9433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-po 5579 df-so 5580 df-cnv 5675 df-iota 6486 df-riota 7358 df-sup 9434 df-inf 9435 |
This theorem is referenced by: xrge0infssd 32469 |
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