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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 3518 and undifdcss 6941 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 4145 |
. . . . 5
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2 | 1 | snid 3638 |
. . . 4
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3 | ssrab2 3255 |
. . . . 5
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4 | p0ex 4203 |
. . . . . . 7
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5 | 4 | rabex 4162 |
. . . . . 6
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6 | sseq12 3195 |
. . . . . . 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 3269 |
. . . . . . . . 9
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10 | 7, 9 | uneq12d 3305 |
. . . . . . . 8
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11 | 10, 8 | eqeq12d 2204 |
. . . . . . 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 2807 |
. . . . 5
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15 | 3, 14 | mpbi 145 |
. . . 4
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16 | 2, 15 | eleqtrri 2265 |
. . 3
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17 | elun 3291 |
. . 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 2909 |
. . . . 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 3273 |
. . . 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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 |
This theorem is referenced by: (None) |
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