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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 4033 | . 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))) | |
2 | nfpo.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2178 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑎 | |
4 | nfpo.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | 3, 4, 3 | nfbr 3808 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑎 |
6 | 5 | nfn 1548 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑎𝑅𝑎 |
7 | nfcv 2178 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑏 | |
8 | 3, 4, 7 | nfbr 3808 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
9 | nfcv 2178 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑐 | |
10 | 7, 4, 9 | nfbr 3808 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐 |
11 | 8, 10 | nfan 1457 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) |
12 | 3, 4, 9 | nfbr 3808 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐 |
13 | 11, 12 | nfim 1464 | . . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐) |
14 | 6, 13 | nfan 1457 | . . . . 5 ⊢ Ⅎ𝑥(¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
15 | 2, 14 | nfralxy 2360 | . . . 4 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
16 | 2, 15 | nfralxy 2360 | . . 3 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
17 | 2, 16 | nfralxy 2360 | . 2 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
18 | 1, 17 | nfxfr 1363 | 1 ⊢ Ⅎ𝑥 𝑅 Po 𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 Ⅎwnf 1349 Ⅎwnfc 2165 ∀wral 2306 class class class wbr 3764 Po wpo 4031 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-po 4033 |
This theorem is referenced by: nfso 4039 |
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