|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > eqinfd | Structured version Visualization version GIF version | ||
| Description: Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020.) | 
| Ref | Expression | 
|---|---|
| infexd.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) | 
| eqinfd.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) | 
| eqinfd.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) | 
| eqinfd.4 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) | 
| Ref | Expression | 
|---|---|
| eqinfd | ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqinfd.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
| 2 | eqinfd.3 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) | |
| 3 | 2 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶) | 
| 4 | eqinfd.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) | |
| 5 | 4 | expr 456 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) | 
| 6 | 5 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) | 
| 7 | infexd.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
| 8 | 7 | eqinf 9524 | . 2 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶)) | 
| 9 | 1, 3, 6, 8 | mp3and 1466 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 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-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: infmin 9534 xrinf0 13380 infmremnf 13385 infmrp1 13386 | 
| Copyright terms: Public domain | W3C validator |