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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 3515 and undifdcss 6935 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 4142 |
. . . . 5
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2 | 1 | snid 3635 |
. . . 4
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3 | ssrab2 3252 |
. . . . 5
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4 | p0ex 4200 |
. . . . . . 7
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5 | 4 | rabex 4159 |
. . . . . 6
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6 | sseq12 3192 |
. . . . . . 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 3266 |
. . . . . . . . 9
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10 | 7, 9 | uneq12d 3302 |
. . . . . . . 8
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11 | 10, 8 | eqeq12d 2202 |
. . . . . . 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 2804 |
. . . . 5
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15 | 3, 14 | mpbi 145 |
. . . 4
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16 | 2, 15 | eleqtrri 2263 |
. . 3
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17 | elun 3288 |
. . 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 2906 |
. . . . 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 3270 |
. . . 4
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24 | 23, 21 | sylnib 677 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 22, 24 | orim12i 760 |
. 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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 |
This theorem is referenced by: (None) |
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