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Mirrors > Home > MPE Home > Th. List > ssnnfi | Structured version Visualization version Unicode version |
Description: A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.) |
Ref | Expression |
---|---|
ssnnfi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspss 3534 |
. . 3
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2 | pssnn 7795 |
. . . . 5
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3 | elnn 6707 |
. . . . . . . . 9
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4 | 3 | expcom 437 |
. . . . . . . 8
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5 | 4 | anim1d 568 |
. . . . . . 7
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6 | 5 | reximdv2 2860 |
. . . . . 6
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7 | 6 | adantr 467 |
. . . . 5
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8 | 2, 7 | mpd 15 |
. . . 4
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9 | eleq1 2519 |
. . . . . 6
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10 | 9 | biimparc 490 |
. . . . 5
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11 | enrefg 7606 |
. . . . . 6
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12 | 11 | ancli 554 |
. . . . 5
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13 | breq2 4409 |
. . . . . 6
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14 | 13 | rspcev 3152 |
. . . . 5
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15 | 10, 12, 14 | 3syl 18 |
. . . 4
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16 | 8, 15 | jaodan 795 |
. . 3
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17 | 1, 16 | sylan2b 478 |
. 2
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18 | isfi 7598 |
. 2
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19 | 17, 18 | sylibr 216 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-sbc 3270 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-om 6698 df-en 7575 df-fin 7578 |
This theorem is referenced by: ssfi 7797 0fin 7804 en1eqsn 7806 isfinite2 7834 pwfi 7874 wofib 8065 infpwfien 8498 fin67 8830 hashcard 12544 rexpen 14292 |
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