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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 5547 | . 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 5551 | . . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴 |
5 | nfcv 2908 | . . . . . . 7 ⊢ Ⅎ𝑥𝑎 | |
6 | nfcv 2908 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
7 | 5, 2, 6 | nfbr 5153 | . . . . . 6 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
8 | nfv 1918 | . . . . . 6 ⊢ Ⅎ𝑥 𝑎 = 𝑏 | |
9 | 6, 2, 5 | nfbr 5153 | . . . . . 6 ⊢ Ⅎ𝑥 𝑏𝑅𝑎 |
10 | 7, 8, 9 | nf3or 1909 | . . . . 5 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) |
11 | 3, 10 | nfralw 3295 | . . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) |
12 | 3, 11 | nfralw 3295 | . . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) |
13 | 4, 12 | nfan 1903 | . 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)) |
14 | 1, 13 | nfxfr 1856 | 1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∨ w3o 1087 Ⅎwnf 1786 Ⅎwnfc 2888 ∀wral 3065 class class class wbr 5106 Po wpo 5544 Or wor 5545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-po 5546 df-so 5547 |
This theorem is referenced by: nfwe 5610 |
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