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Mirrors > Home > MPE Home > Th. List > isfin1-2 | Structured version Unicode version |
Description: A set is finite in the usual sense iff the power set of its power set is Dedekind finite. (Contributed by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
isfin1-2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3081 |
. 2
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2 | elex 3081 |
. . 3
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3 | pwexb 6492 |
. . . 4
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4 | pwexb 6492 |
. . . 4
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5 | 3, 4 | bitri 249 |
. . 3
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6 | 2, 5 | sylibr 212 |
. 2
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7 | ominf 7631 |
. . . . . 6
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8 | pwfi 7712 |
. . . . . . . 8
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9 | pwfi 7712 |
. . . . . . . 8
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10 | 8, 9 | bitri 249 |
. . . . . . 7
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11 | domfi 7640 |
. . . . . . . 8
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12 | 11 | expcom 435 |
. . . . . . 7
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13 | 10, 12 | syl5bi 217 |
. . . . . 6
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14 | 7, 13 | mtoi 178 |
. . . . 5
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15 | fineqvlem 7633 |
. . . . . 6
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16 | 15 | ex 434 |
. . . . 5
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17 | 14, 16 | impbid2 204 |
. . . 4
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18 | 17 | con2bid 329 |
. . 3
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19 | isfin4-2 8589 |
. . . 4
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20 | 5, 19 | sylbi 195 |
. . 3
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21 | 18, 20 | bitr4d 256 |
. 2
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22 | 1, 6, 21 | pm5.21nii 353 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 ax-rep 4506 ax-sep 4516 ax-nul 4524 ax-pow 4573 ax-pr 4634 ax-un 6477 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-ral 2801 df-rex 2802 df-reu 2803 df-rab 2805 df-v 3074 df-sbc 3289 df-csb 3391 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-pss 3447 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4195 df-int 4232 df-iun 4276 df-br 4396 df-opab 4454 df-mpt 4455 df-tr 4489 df-eprel 4735 df-id 4739 df-po 4744 df-so 4745 df-fr 4782 df-we 4784 df-ord 4825 df-on 4826 df-lim 4827 df-suc 4828 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-rn 4954 df-res 4955 df-ima 4956 df-iota 5484 df-fun 5523 df-fn 5524 df-f 5525 df-f1 5526 df-fo 5527 df-f1o 5528 df-fv 5529 df-ov 6198 df-oprab 6199 df-mpt2 6200 df-om 6582 df-1st 6682 df-2nd 6683 df-recs 6937 df-rdg 6971 df-1o 7025 df-2o 7026 df-oadd 7029 df-er 7206 df-map 7321 df-en 7416 df-dom 7417 df-sdom 7418 df-fin 7419 df-fin4 8562 |
This theorem is referenced by: (None) |
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