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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 9528 | . 2 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴) | 
| 4 | infssd.1 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
| 5 | 4 | sseld 3982 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵)) | 
| 6 | 1, 2 | inflb 9529 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐵 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅))) | 
| 7 | 5, 6 | syld 47 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐶 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅))) | 
| 8 | 7 | ralrimiv 3145 | . 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 9532 | . 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 699 | 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 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 Or wor 5591 infcinf 9481 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-po 5592 df-so 5593 df-cnv 5693 df-iota 6514 df-riota 7388 df-sup 9482 df-inf 9483 | 
| This theorem is referenced by: xrge0infssd 32765 | 
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