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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 3095 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶) |
4 | eqinfd.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) | |
5 | 4 | expr 460 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) |
6 | 5 | ralrimiva 3095 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) |
7 | infexd.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
8 | 7 | eqinf 9078 | . 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 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 class class class wbr 5039 Or wor 5452 infcinf 9035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-po 5453 df-so 5454 df-cnv 5544 df-iota 6316 df-riota 7148 df-sup 9036 df-inf 9037 |
This theorem is referenced by: infmin 9088 xrinf0 12893 infmremnf 12898 infmrp1 12899 |
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