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Mirrors > Home > ILE Home > Th. List > nfsup | GIF version |
Description: Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.) |
Ref | Expression |
---|---|
nfsup.1 | ⊢ Ⅎ𝑥𝐴 |
nfsup.2 | ⊢ Ⅎ𝑥𝐵 |
nfsup.3 | ⊢ Ⅎ𝑥𝑅 |
Ref | Expression |
---|---|
nfsup | ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sup 6397 | . 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} | |
2 | nfsup.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2219 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑢 | |
4 | nfsup.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2219 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑣 | |
6 | 3, 4, 5 | nfbr 3829 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑢𝑅𝑣 |
7 | 6 | nfn 1588 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣 |
8 | 2, 7 | nfralya 2404 | . . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 |
9 | nfsup.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
10 | 5, 4, 3 | nfbr 3829 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢 |
11 | nfcv 2219 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑤 | |
12 | 5, 4, 11 | nfbr 3829 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑣𝑅𝑤 |
13 | 2, 12 | nfrexya 2405 | . . . . . . 7 ⊢ Ⅎ𝑥∃𝑤 ∈ 𝐴 𝑣𝑅𝑤 |
14 | 10, 13 | nfim 1504 | . . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤) |
15 | 9, 14 | nfralya 2404 | . . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤) |
16 | 8, 15 | nfan 1497 | . . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤)) |
17 | 16, 9 | nfrabxy 2534 | . . 3 ⊢ Ⅎ𝑥{𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} |
18 | 17 | nfuni 3607 | . 2 ⊢ Ⅎ𝑥∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} |
19 | 1, 18 | nfcxfr 2216 | 1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 Ⅎwnfc 2206 ∀wral 2348 ∃wrex 2349 {crab 2352 ∪ cuni 3601 class class class wbr 3785 supcsup 6395 |
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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-sup 6397 |
This theorem is referenced by: nfinf 6430 infssuzcldc 10347 |
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