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Mirrors > Home > ILE Home > Th. List > nfpo | GIF version |
Description: Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
Ref | Expression |
---|---|
nfpo.r | ⊢ Ⅎ𝑥𝑅 |
nfpo.a | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfpo | ⊢ Ⅎ𝑥 𝑅 Po 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-po 4051 | . 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))) | |
2 | nfpo.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2219 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑎 | |
4 | nfpo.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | 3, 4, 3 | nfbr 3829 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑎 |
6 | 5 | nfn 1588 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑎𝑅𝑎 |
7 | nfcv 2219 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑏 | |
8 | 3, 4, 7 | nfbr 3829 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
9 | nfcv 2219 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑐 | |
10 | 7, 4, 9 | nfbr 3829 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐 |
11 | 8, 10 | nfan 1497 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) |
12 | 3, 4, 9 | nfbr 3829 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐 |
13 | 11, 12 | nfim 1504 | . . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐) |
14 | 6, 13 | nfan 1497 | . . . . 5 ⊢ Ⅎ𝑥(¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
15 | 2, 14 | nfralxy 2402 | . . . 4 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
16 | 2, 15 | nfralxy 2402 | . . 3 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
17 | 2, 16 | nfralxy 2402 | . 2 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
18 | 1, 17 | nfxfr 1403 | 1 ⊢ Ⅎ𝑥 𝑅 Po 𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 Ⅎwnf 1389 Ⅎwnfc 2206 ∀wral 2348 class class class wbr 3785 Po wpo 4049 |
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-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-po 4051 |
This theorem is referenced by: nfso 4057 |
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