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Mirrors > Home > ILE Home > Th. List > nfso | GIF version |
Description: Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
Ref | Expression |
---|---|
nfpo.r | ⊢ Ⅎ𝑥𝑅 |
nfpo.a | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfso | ⊢ Ⅎ𝑥 𝑅 Or 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iso 4034 | . 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)))) | |
2 | nfpo.r | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
3 | nfpo.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfpo 4038 | . . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴 |
5 | nfcv 2178 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑎 | |
6 | nfcv 2178 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑏 | |
7 | 5, 2, 6 | nfbr 3808 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
8 | nfcv 2178 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑐 | |
9 | 5, 2, 8 | nfbr 3808 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑐 |
10 | 8, 2, 6 | nfbr 3808 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑐𝑅𝑏 |
11 | 9, 10 | nfor 1466 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏) |
12 | 7, 11 | nfim 1464 | . . . . . 6 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)) |
13 | 3, 12 | nfralxy 2360 | . . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)) |
14 | 3, 13 | nfralxy 2360 | . . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)) |
15 | 3, 14 | nfralxy 2360 | . . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)) |
16 | 4, 15 | nfan 1457 | . 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))) |
17 | 1, 16 | nfxfr 1363 | 1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∨ wo 629 Ⅎwnf 1349 Ⅎwnfc 2165 ∀wral 2306 class class class wbr 3764 Po wpo 4031 Or wor 4032 |
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 df-iso 4034 |
This theorem is referenced by: (None) |
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