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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 7001 | . 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} | |
2 | nfsup.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2332 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑢 | |
4 | nfsup.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2332 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑣 | |
6 | 3, 4, 5 | nfbr 4064 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑢𝑅𝑣 |
7 | 6 | nfn 1669 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣 |
8 | 2, 7 | nfralya 2530 | . . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 |
9 | nfsup.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
10 | 5, 4, 3 | nfbr 4064 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢 |
11 | nfcv 2332 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑤 | |
12 | 5, 4, 11 | nfbr 4064 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑣𝑅𝑤 |
13 | 2, 12 | nfrexya 2531 | . . . . . . 7 ⊢ Ⅎ𝑥∃𝑤 ∈ 𝐴 𝑣𝑅𝑤 |
14 | 10, 13 | nfim 1583 | . . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤) |
15 | 9, 14 | nfralya 2530 | . . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤) |
16 | 8, 15 | nfan 1576 | . . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤)) |
17 | 16, 9 | nfrabxy 2671 | . . 3 ⊢ Ⅎ𝑥{𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} |
18 | 17 | nfuni 3830 | . 2 ⊢ Ⅎ𝑥∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} |
19 | 1, 18 | nfcxfr 2329 | 1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 Ⅎwnfc 2319 ∀wral 2468 ∃wrex 2469 {crab 2472 ∪ cuni 3824 class class class wbr 4018 supcsup 6999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-sup 7001 |
This theorem is referenced by: nfinf 7034 infssuzcldc 11970 |
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