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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 6839 | . 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} | |
2 | nfsup.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2258 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑢 | |
4 | nfsup.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2258 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑣 | |
6 | 3, 4, 5 | nfbr 3944 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑢𝑅𝑣 |
7 | 6 | nfn 1621 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣 |
8 | 2, 7 | nfralya 2450 | . . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 |
9 | nfsup.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
10 | 5, 4, 3 | nfbr 3944 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢 |
11 | nfcv 2258 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑤 | |
12 | 5, 4, 11 | nfbr 3944 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑣𝑅𝑤 |
13 | 2, 12 | nfrexya 2451 | . . . . . . 7 ⊢ Ⅎ𝑥∃𝑤 ∈ 𝐴 𝑣𝑅𝑤 |
14 | 10, 13 | nfim 1536 | . . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤) |
15 | 9, 14 | nfralya 2450 | . . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤) |
16 | 8, 15 | nfan 1529 | . . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤)) |
17 | 16, 9 | nfrabxy 2588 | . . 3 ⊢ Ⅎ𝑥{𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} |
18 | 17 | nfuni 3712 | . 2 ⊢ Ⅎ𝑥∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} |
19 | 1, 18 | nfcxfr 2255 | 1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 Ⅎwnfc 2245 ∀wral 2393 ∃wrex 2394 {crab 2397 ∪ cuni 3706 class class class wbr 3899 supcsup 6837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-sup 6839 |
This theorem is referenced by: nfinf 6872 infssuzcldc 11571 |
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