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Mirrors > Home > MPE Home > Th. List > nfso | Structured version Visualization version 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-so 5546 | . 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))) | |
2 | nfpo.r | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
3 | nfpo.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfpo 5550 | . . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴 |
5 | nfcv 2907 | . . . . . . 7 ⊢ Ⅎ𝑥𝑎 | |
6 | nfcv 2907 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
7 | 5, 2, 6 | nfbr 5152 | . . . . . 6 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
8 | nfv 1917 | . . . . . 6 ⊢ Ⅎ𝑥 𝑎 = 𝑏 | |
9 | 6, 2, 5 | nfbr 5152 | . . . . . 6 ⊢ Ⅎ𝑥 𝑏𝑅𝑎 |
10 | 7, 8, 9 | nf3or 1908 | . . . . 5 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) |
11 | 3, 10 | nfralw 3294 | . . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) |
12 | 3, 11 | nfralw 3294 | . . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) |
13 | 4, 12 | nfan 1902 | . 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)) |
14 | 1, 13 | nfxfr 1855 | 1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∨ w3o 1086 Ⅎwnf 1785 Ⅎwnfc 2887 ∀wral 3064 class class class wbr 5105 Po wpo 5543 Or wor 5544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ral 3065 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-po 5545 df-so 5546 |
This theorem is referenced by: nfwe 5609 |
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