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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 3505 and undifdcss 6925 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 4132 |
. . . . 5
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2 | 1 | snid 3625 |
. . . 4
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3 | ssrab2 3242 |
. . . . 5
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4 | p0ex 4190 |
. . . . . . 7
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5 | 4 | rabex 4149 |
. . . . . 6
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6 | sseq12 3182 |
. . . . . . 7
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7 | simpl 109 |
. . . . . . . . 9
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8 | simpr 110 |
. . . . . . . . . 10
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9 | 8, 7 | difeq12d 3256 |
. . . . . . . . 9
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10 | 7, 9 | uneq12d 3292 |
. . . . . . . 8
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11 | 10, 8 | eqeq12d 2192 |
. . . . . . 7
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12 | 6, 11 | bibi12d 235 |
. . . . . 6
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13 | undifexmid.1 |
. . . . . 6
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14 | 5, 4, 12, 13 | vtocl2 2794 |
. . . . 5
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15 | 3, 14 | mpbi 145 |
. . . 4
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16 | 2, 15 | eleqtrri 2253 |
. . 3
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17 | elun 3278 |
. . 3
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18 | 16, 17 | mpbi 145 |
. 2
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19 | biidd 172 |
. . . . . 6
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20 | 19 | elrab3 2896 |
. . . . 5
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21 | 2, 20 | ax-mp 5 |
. . . 4
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22 | 21 | biimpi 120 |
. . 3
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23 | eldifn 3260 |
. . . 4
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24 | 23, 21 | sylnib 676 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 22, 24 | orim12i 759 |
. 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 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 |
This theorem is referenced by: (None) |
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