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Mirrors > Home > ILE Home > Th. List > domfiexmid | Unicode version |
Description: If any set dominated by a finite set is finite, excluded middle follows. (Contributed by Jim Kingdon, 3-Feb-2022.) |
Ref | Expression |
---|---|
domfiexmid.1 |
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Ref | Expression |
---|---|
domfiexmid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4148 |
. . . 4
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2 | snfig 6844 |
. . . 4
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3 | 1, 2 | ax-mp 5 |
. . 3
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4 | ssrab2 3255 |
. . . 4
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5 | ssdomg 6808 |
. . . 4
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6 | 3, 4, 5 | mp2 16 |
. . 3
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7 | domfiexmid.1 |
. . . . . 6
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8 | 7 | gen2 1461 |
. . . . 5
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9 | p0ex 4209 |
. . . . . 6
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10 | eleq1 2252 |
. . . . . . . . 9
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11 | breq2 4025 |
. . . . . . . . 9
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12 | 10, 11 | anbi12d 473 |
. . . . . . . 8
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13 | 12 | imbi1d 231 |
. . . . . . 7
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14 | 13 | albidv 1835 |
. . . . . 6
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15 | 9, 14 | spcv 2846 |
. . . . 5
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16 | 8, 15 | ax-mp 5 |
. . . 4
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17 | 9 | rabex 4165 |
. . . . 5
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18 | breq1 4024 |
. . . . . . 7
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19 | 18 | anbi2d 464 |
. . . . . 6
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20 | eleq1 2252 |
. . . . . 6
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21 | 19, 20 | imbi12d 234 |
. . . . 5
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22 | 17, 21 | spcv 2846 |
. . . 4
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23 | 16, 22 | ax-mp 5 |
. . 3
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24 | 3, 6, 23 | mp2an 426 |
. 2
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25 | 24 | ssfilem 6907 |
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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-iinf 4608 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-br 4022 df-opab 4083 df-id 4314 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-1o 6445 df-er 6563 df-en 6771 df-dom 6772 df-fin 6773 |
This theorem is referenced by: (None) |
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