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Mirrors > Home > MPE Home > Th. List > rankr1ai | Structured version Visualization version Unicode version |
Description: One direction of rankr1a 8699. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankr1ai |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6220 | . . 3 | |
2 | r1val1 8649 | . . . . . 6 | |
3 | 2 | eleq2d 2687 | . . . . 5 |
4 | eliun 4524 | . . . . 5 | |
5 | 3, 4 | syl6bb 276 | . . . 4 |
6 | r1funlim 8629 | . . . . . . . . . . 11 | |
7 | 6 | simpri 478 | . . . . . . . . . 10 |
8 | limord 5784 | . . . . . . . . . 10 | |
9 | 7, 8 | ax-mp 5 | . . . . . . . . 9 |
10 | ordtr1 5767 | . . . . . . . . 9 | |
11 | 9, 10 | ax-mp 5 | . . . . . . . 8 |
12 | 11 | ancoms 469 | . . . . . . 7 |
13 | r1sucg 8632 | . . . . . . . 8 | |
14 | 13 | eleq2d 2687 | . . . . . . 7 |
15 | 12, 14 | syl 17 | . . . . . 6 |
16 | ordsson 6989 | . . . . . . . . . 10 | |
17 | 9, 16 | ax-mp 5 | . . . . . . . . 9 |
18 | 17, 12 | sseldi 3601 | . . . . . . . 8 |
19 | rabid 3116 | . . . . . . . . 9 | |
20 | intss1 4492 | . . . . . . . . 9 | |
21 | 19, 20 | sylbir 225 | . . . . . . . 8 |
22 | 18, 21 | sylan 488 | . . . . . . 7 |
23 | 22 | ex 450 | . . . . . 6 |
24 | 15, 23 | sylbird 250 | . . . . 5 |
25 | 24 | reximdva 3017 | . . . 4 |
26 | 5, 25 | sylbid 230 | . . 3 |
27 | 1, 26 | mpcom 38 | . 2 |
28 | r1elwf 8659 | . . . . . . 7 | |
29 | rankvalb 8660 | . . . . . . 7 | |
30 | 28, 29 | syl 17 | . . . . . 6 |
31 | 30 | sseq1d 3632 | . . . . 5 |
32 | 31 | adantr 481 | . . . 4 |
33 | rankon 8658 | . . . . . . 7 | |
34 | 17, 1 | sseldi 3601 | . . . . . . 7 |
35 | ontr2 5772 | . . . . . . 7 | |
36 | 33, 34, 35 | sylancr 695 | . . . . . 6 |
37 | 36 | expcomd 454 | . . . . 5 |
38 | 37 | imp 445 | . . . 4 |
39 | 32, 38 | sylbird 250 | . . 3 |
40 | 39 | rexlimdva 3031 | . 2 |
41 | 27, 40 | mpd 15 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wrex 2913 crab 2916 wss 3574 cpw 4158 cuni 4436 cint 4475 ciun 4520 cdm 5114 cima 5117 word 5722 con0 5723 wlim 5724 csuc 5725 wfun 5882 cfv 5888 cr1 8625 crnk 8626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-r1 8627 df-rank 8628 |
This theorem is referenced by: rankr1ag 8665 tcrank 8747 dfac12lem1 8965 dfac12lem2 8966 r1limwun 9558 inatsk 9600 aomclem4 37627 |
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