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Mirrors > Home > MPE Home > Th. List > isfin3ds | Structured version Visualization version Unicode version |
Description: Property of a III-finite set (descending sequence version). (Contributed by Mario Carneiro, 16-May-2015.) |
Ref | Expression |
---|---|
isfin3ds.f |
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Ref | Expression |
---|---|
isfin3ds |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suceq 5495 |
. . . . . . . . 9
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2 | 1 | fveq2d 5883 |
. . . . . . . 8
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3 | fveq2 5879 |
. . . . . . . 8
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4 | 2, 3 | sseq12d 3447 |
. . . . . . 7
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5 | 4 | cbvralv 3005 |
. . . . . 6
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6 | fveq1 5878 |
. . . . . . . 8
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7 | fveq1 5878 |
. . . . . . . 8
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8 | 6, 7 | sseq12d 3447 |
. . . . . . 7
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9 | 8 | ralbidv 2829 |
. . . . . 6
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10 | 5, 9 | syl5bb 265 |
. . . . 5
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11 | rneq 5066 |
. . . . . . 7
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12 | 11 | inteqd 4231 |
. . . . . 6
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13 | 12, 11 | eleq12d 2543 |
. . . . 5
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14 | 10, 13 | imbi12d 327 |
. . . 4
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15 | 14 | cbvralv 3005 |
. . 3
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16 | pweq 3945 |
. . . . 5
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17 | 16 | oveq1d 6323 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17 | raleqdv 2979 |
. . 3
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19 | 15, 18 | syl5bb 265 |
. 2
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20 | isfin3ds.f |
. 2
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21 | 19, 20 | elab2g 3175 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-int 4227 df-br 4396 df-opab 4455 df-cnv 4847 df-dm 4849 df-rn 4850 df-suc 5436 df-iota 5553 df-fv 5597 df-ov 6311 |
This theorem is referenced by: ssfin3ds 8778 fin23lem17 8786 fin23lem39 8798 fin23lem40 8799 isf32lem12 8812 isfin3-3 8816 |
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