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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 5139 | . 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))) | |
2 | nfpo.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2866 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑎 | |
4 | nfpo.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | 3, 4, 3 | nfbr 4807 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑎 |
6 | 5 | nfn 1897 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑎𝑅𝑎 |
7 | nfcv 2866 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑏 | |
8 | 3, 4, 7 | nfbr 4807 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
9 | nfcv 2866 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑐 | |
10 | 7, 4, 9 | nfbr 4807 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐 |
11 | 8, 10 | nfan 1941 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) |
12 | 3, 4, 9 | nfbr 4807 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐 |
13 | 11, 12 | nfim 1938 | . . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐) |
14 | 6, 13 | nfan 1941 | . . . . 5 ⊢ Ⅎ𝑥(¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
15 | 2, 14 | nfral 3047 | . . . 4 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
16 | 2, 15 | nfral 3047 | . . 3 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
17 | 2, 16 | nfral 3047 | . 2 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
18 | 1, 17 | nfxfr 1892 | 1 ⊢ Ⅎ𝑥 𝑅 Po 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 Ⅎwnf 1821 Ⅎwnfc 2853 ∀wral 3014 class class class wbr 4760 Po wpo 5137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ral 3019 df-rab 3023 df-v 3306 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-nul 4024 df-if 4195 df-sn 4286 df-pr 4288 df-op 4292 df-br 4761 df-po 5139 |
This theorem is referenced by: nfso 5145 |
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