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Mirrors > Home > ILE Home > Th. List > eqinftid | GIF version |
Description: Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.) |
Ref | Expression |
---|---|
eqinfti.ti | ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) |
eqinftid.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
eqinftid.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) |
eqinftid.4 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
Ref | Expression |
---|---|
eqinftid | ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqinftid.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
2 | eqinftid.3 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) | |
3 | 2 | ralrimiva 2434 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶) |
4 | eqinftid.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) | |
5 | 4 | expr 367 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) |
6 | 5 | ralrimiva 2434 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) |
7 | eqinfti.ti | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) | |
8 | 7 | eqinfti 6433 | . 2 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶)) |
9 | 1, 3, 6, 8 | mp3and 1271 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 class class class wbr 3785 infcinf 6396 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-cnv 4371 df-iota 4887 df-riota 5488 df-sup 6397 df-inf 6398 |
This theorem is referenced by: infminti 6440 |
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