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Mirrors > Home > MPE Home > Th. List > fin23lem31 | Structured version Visualization version Unicode version |
Description: Lemma for fin23 8816. The residual is has a strictly smaller range than the previous sequence. This will be iterated to build an unbounded chain. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
fin23lem.a |
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fin23lem17.f |
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fin23lem.b |
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fin23lem.c |
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fin23lem.d |
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fin23lem.e |
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Ref | Expression |
---|---|
fin23lem31 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fin23lem17.f |
. . . 4
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2 | 1 | ssfin3ds 8757 |
. . 3
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3 | fin23lem.a |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | fin23lem.b |
. . . . . 6
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5 | fin23lem.c |
. . . . . 6
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6 | fin23lem.d |
. . . . . 6
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7 | fin23lem.e |
. . . . . 6
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8 | 3, 1, 4, 5, 6, 7 | fin23lem29 8768 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 8 | a1i 11 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 3, 1 | fin23lem21 8766 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 10 | ancoms 455 |
. . . . . 6
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12 | n0 3740 |
. . . . . 6
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13 | 11, 12 | sylib 200 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 3 | fnseqom 7169 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() |
15 | fndm 5673 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 14, 15 | ax-mp 5 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() |
17 | peano1 6709 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() | |
18 | 17 | ne0ii 3737 |
. . . . . . . . . . . . 13
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19 | 16, 18 | eqnetri 2693 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() |
20 | dm0rn0 5050 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 20 | necon3bii 2675 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 19, 21 | mpbi 212 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() |
23 | intssuni 4256 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | ax-mp 5 |
. . . . . . . . . 10
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25 | 3 | fin23lem16 8762 |
. . . . . . . . . 10
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26 | 24, 25 | sseqtri 3463 |
. . . . . . . . 9
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27 | 26 | sseli 3427 |
. . . . . . . 8
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28 | 27 | adantl 468 |
. . . . . . 7
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29 | f1fun 5779 |
. . . . . . . . . . . . 13
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30 | 29 | adantr 467 |
. . . . . . . . . . . 12
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31 | 3, 1, 4, 5, 6, 7 | fin23lem30 8769 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 30, 31 | syl 17 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | disj 3804 |
. . . . . . . . . . 11
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34 | 32, 33 | sylib 200 |
. . . . . . . . . 10
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35 | rsp 2753 |
. . . . . . . . . 10
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36 | 34, 35 | syl 17 |
. . . . . . . . 9
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37 | 36 | con2d 119 |
. . . . . . . 8
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38 | 37 | imp 431 |
. . . . . . 7
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39 | nelne1 2719 |
. . . . . . 7
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40 | 28, 38, 39 | syl2anc 666 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 40 | necomd 2678 |
. . . . 5
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42 | 13, 41 | exlimddv 1780 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | df-pss 3419 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
44 | 9, 42, 43 | sylanbrc 669 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 2, 44 | sylan2 477 |
. 2
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46 | 45 | 3impb 1203 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-int 4234 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-se 4793 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-isom 5590 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-om 6690 df-1st 6790 df-2nd 6791 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-seqom 7162 df-1o 7179 df-oadd 7183 df-er 7360 df-map 7471 df-en 7567 df-dom 7568 df-sdom 7569 df-fin 7570 df-card 8370 |
This theorem is referenced by: fin23lem32 8771 |
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