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Mirrors > Home > MPE Home > Th. List > fin1a2lem4 | Structured version Visualization version Unicode version |
Description: Lemma for fin1a2 8850. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Ref | Expression |
---|---|
fin1a2lem.b |
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Ref | Expression |
---|---|
fin1a2lem4 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fin1a2lem.b |
. . 3
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2 | 2onn 7346 |
. . . 4
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3 | nnmcl 7318 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 2, 3 | mpan 677 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 1, 4 | fmpti 6050 |
. 2
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6 | 1 | fin1a2lem3 8837 |
. . . . . 6
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7 | 1 | fin1a2lem3 8837 |
. . . . . 6
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8 | 6, 7 | eqeqan12d 2469 |
. . . . 5
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9 | 2on 7195 |
. . . . . . 7
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10 | 9 | a1i 11 |
. . . . . 6
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11 | nnon 6703 |
. . . . . . 7
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12 | 11 | adantr 467 |
. . . . . 6
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13 | nnon 6703 |
. . . . . . 7
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14 | 13 | adantl 468 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 0lt1o 7211 |
. . . . . . . . 9
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16 | elelsuc 5498 |
. . . . . . . . 9
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17 | 15, 16 | ax-mp 5 |
. . . . . . . 8
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18 | df-2o 7188 |
. . . . . . . 8
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19 | 17, 18 | eleqtrri 2530 |
. . . . . . 7
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20 | 19 | a1i 11 |
. . . . . 6
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21 | omcan 7275 |
. . . . . 6
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22 | 10, 12, 14, 20, 21 | syl31anc 1272 |
. . . . 5
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23 | 8, 22 | bitrd 257 |
. . . 4
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24 | 23 | biimpd 211 |
. . 3
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25 | 24 | rgen2a 2817 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | dff13 6164 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 5, 25, 26 | mpbir2an 932 |
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-rep 4518 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-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 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-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 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-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-1o 7187 df-2o 7188 df-oadd 7191 df-omul 7192 |
This theorem is referenced by: fin1a2lem5 8839 fin1a2lem6 8840 fin1a2lem7 8841 |
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