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Mirrors > Home > ILE Home > Th. List > undifexmid | Unicode version |
Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3448 and undifdcss 6819 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.) |
Ref | Expression |
---|---|
undifexmid.1 |
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Ref | Expression |
---|---|
undifexmid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4063 |
. . . . 5
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2 | 1 | snid 3563 |
. . . 4
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3 | ssrab2 3187 |
. . . . 5
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4 | p0ex 4120 |
. . . . . . 7
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5 | 4 | rabex 4080 |
. . . . . 6
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6 | sseq12 3127 |
. . . . . . 7
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7 | simpl 108 |
. . . . . . . . 9
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8 | simpr 109 |
. . . . . . . . . 10
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9 | 8, 7 | difeq12d 3200 |
. . . . . . . . 9
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10 | 7, 9 | uneq12d 3236 |
. . . . . . . 8
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11 | 10, 8 | eqeq12d 2155 |
. . . . . . 7
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12 | 6, 11 | bibi12d 234 |
. . . . . 6
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13 | undifexmid.1 |
. . . . . 6
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14 | 5, 4, 12, 13 | vtocl2 2744 |
. . . . 5
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15 | 3, 14 | mpbi 144 |
. . . 4
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16 | 2, 15 | eleqtrri 2216 |
. . 3
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17 | elun 3222 |
. . 3
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18 | 16, 17 | mpbi 144 |
. 2
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19 | biidd 171 |
. . . . . 6
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20 | 19 | elrab3 2845 |
. . . . 5
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21 | 2, 20 | ax-mp 5 |
. . . 4
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22 | 21 | biimpi 119 |
. . 3
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23 | eldifn 3204 |
. . . 4
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24 | 23, 21 | sylnib 666 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 22, 24 | orim12i 749 |
. 2
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26 | 18, 25 | ax-mp 5 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 |
This theorem is referenced by: (None) |
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