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Mirrors > Home > ILE Home > Th. List > ssfiexmid | Unicode version |
Description: If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.) |
Ref | Expression |
---|---|
ssfiexmid.1 |
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Ref | Expression |
---|---|
ssfiexmid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 3966 |
. . . 4
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2 | snfig 6529 |
. . . 4
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3 | 1, 2 | ax-mp 7 |
. . 3
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4 | ssrab2 3106 |
. . 3
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5 | ssfiexmid.1 |
. . . . 5
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6 | p0ex 4023 |
. . . . . 6
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7 | eleq1 2150 |
. . . . . . . . 9
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8 | sseq2 3048 |
. . . . . . . . 9
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9 | 7, 8 | anbi12d 457 |
. . . . . . . 8
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10 | 9 | imbi1d 229 |
. . . . . . 7
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11 | 10 | albidv 1752 |
. . . . . 6
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12 | 6, 11 | spcv 2712 |
. . . . 5
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13 | 5, 12 | ax-mp 7 |
. . . 4
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14 | 6 | rabex 3983 |
. . . . 5
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15 | sseq1 3047 |
. . . . . . 7
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16 | 15 | anbi2d 452 |
. . . . . 6
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17 | eleq1 2150 |
. . . . . 6
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18 | 16, 17 | imbi12d 232 |
. . . . 5
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19 | 14, 18 | spcv 2712 |
. . . 4
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20 | 13, 19 | ax-mp 7 |
. . 3
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21 | 3, 4, 20 | mp2an 417 |
. 2
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22 | 21 | ssfilem 6589 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-br 3846 df-opab 3900 df-id 4120 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-1o 6181 df-er 6290 df-en 6456 df-fin 6458 |
This theorem is referenced by: infiexmid 6591 |
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