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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 6957 | . 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} | |
2 | nfsup.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2312 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑢 | |
4 | nfsup.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2312 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑣 | |
6 | 3, 4, 5 | nfbr 4033 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑢𝑅𝑣 |
7 | 6 | nfn 1651 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣 |
8 | 2, 7 | nfralya 2510 | . . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 |
9 | nfsup.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
10 | 5, 4, 3 | nfbr 4033 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢 |
11 | nfcv 2312 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑤 | |
12 | 5, 4, 11 | nfbr 4033 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑣𝑅𝑤 |
13 | 2, 12 | nfrexya 2511 | . . . . . . 7 ⊢ Ⅎ𝑥∃𝑤 ∈ 𝐴 𝑣𝑅𝑤 |
14 | 10, 13 | nfim 1565 | . . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤) |
15 | 9, 14 | nfralya 2510 | . . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤) |
16 | 8, 15 | nfan 1558 | . . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤)) |
17 | 16, 9 | nfrabxy 2650 | . . 3 ⊢ Ⅎ𝑥{𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} |
18 | 17 | nfuni 3800 | . 2 ⊢ Ⅎ𝑥∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))} |
19 | 1, 18 | nfcxfr 2309 | 1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 Ⅎwnfc 2299 ∀wral 2448 ∃wrex 2449 {crab 2452 ∪ cuni 3794 class class class wbr 3987 supcsup 6955 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-sup 6957 |
This theorem is referenced by: nfinf 6990 infssuzcldc 11893 |
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