Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfpo | Structured version Visualization version 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 5468 | . 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))) | |
2 | nfpo.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑎 | |
4 | nfpo.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | 3, 4, 3 | nfbr 5100 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑎 |
6 | 5 | nfn 1865 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑎𝑅𝑎 |
7 | nfcv 2904 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑏 | |
8 | 3, 4, 7 | nfbr 5100 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
9 | nfcv 2904 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑐 | |
10 | 7, 4, 9 | nfbr 5100 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐 |
11 | 8, 10 | nfan 1907 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) |
12 | 3, 4, 9 | nfbr 5100 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐 |
13 | 11, 12 | nfim 1904 | . . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐) |
14 | 6, 13 | nfan 1907 | . . . . 5 ⊢ Ⅎ𝑥(¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
15 | 2, 14 | nfralw 3147 | . . . 4 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
16 | 2, 15 | nfralw 3147 | . . 3 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
17 | 2, 16 | nfralw 3147 | . 2 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
18 | 1, 17 | nfxfr 1860 | 1 ⊢ Ⅎ𝑥 𝑅 Po 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 Ⅎwnf 1791 Ⅎwnfc 2884 ∀wral 3061 class class class wbr 5053 Po wpo 5466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-po 5468 |
This theorem is referenced by: nfso 5474 |
Copyright terms: Public domain | W3C validator |