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Mirrors > Home > MPE Home > Th. List > ssnnfi | Structured 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 3566 |
. . 3
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2 | pssnn 7645 |
. . . . 5
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3 | elnn 6599 |
. . . . . . . . 9
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4 | 3 | expcom 435 |
. . . . . . . 8
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5 | 4 | anim1d 564 |
. . . . . . 7
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6 | 5 | reximdv2 2931 |
. . . . . 6
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7 | 6 | adantr 465 |
. . . . 5
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8 | 2, 7 | mpd 15 |
. . . 4
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9 | eleq1 2526 |
. . . . . 6
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10 | 9 | biimparc 487 |
. . . . 5
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11 | enrefg 7454 |
. . . . . 6
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12 | 11 | ancli 551 |
. . . . 5
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13 | breq2 4407 |
. . . . . 6
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14 | 13 | rspcev 3179 |
. . . . 5
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15 | 10, 12, 14 | 3syl 20 |
. . . 4
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16 | 8, 15 | jaodan 783 |
. . 3
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17 | 1, 16 | sylan2b 475 |
. 2
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18 | isfi 7446 |
. 2
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19 | 17, 18 | sylibr 212 |
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 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
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 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-br 4404 df-opab 4462 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-om 6590 df-en 7424 df-fin 7427 |
This theorem is referenced by: ssfi 7647 0fin 7654 en1eqsn 7656 isfinite2 7684 pwfi 7720 wofib 7874 infpwfien 8347 fin67 8679 hashcard 12246 rexpen 13632 |
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