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Mirrors > Home > MPE Home > Th. List > canthnumlem | Structured version Visualization version Unicode version |
Description: Lemma for canthnum 9471. (Contributed by Mario Carneiro, 19-May-2015.) |
Ref | Expression |
---|---|
canth4.1 | |
canth4.2 | |
canth4.3 |
Ref | Expression |
---|---|
canthnumlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6101 | . . . . 5 | |
2 | ssid 3624 | . . . . . 6 | |
3 | canth4.1 | . . . . . . 7 | |
4 | canth4.2 | . . . . . . 7 | |
5 | canth4.3 | . . . . . . 7 | |
6 | 3, 4, 5 | canth4 9469 | . . . . . 6 |
7 | 2, 6 | mp3an3 1413 | . . . . 5 |
8 | 1, 7 | sylan2 491 | . . . 4 |
9 | 8 | simp3d 1075 | . . 3 |
10 | simpr 477 | . . . 4 | |
11 | 8 | simp1d 1073 | . . . . . 6 |
12 | elpw2g 4827 | . . . . . . 7 | |
13 | 12 | adantr 481 | . . . . . 6 |
14 | 11, 13 | mpbird 247 | . . . . 5 |
15 | eqid 2622 | . . . . . . . . . . . . 13 | |
16 | eqid 2622 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | pm3.2i 471 | . . . . . . . . . . . 12 |
18 | elex 3212 | . . . . . . . . . . . . . 14 | |
19 | 18 | adantr 481 | . . . . . . . . . . . . 13 |
20 | 10, 1 | syl 17 | . . . . . . . . . . . . . 14 |
21 | 20 | ffvelrnda 6359 | . . . . . . . . . . . . 13 |
22 | 3, 19, 21, 4 | fpwwe 9468 | . . . . . . . . . . . 12 |
23 | 17, 22 | mpbiri 248 | . . . . . . . . . . 11 |
24 | 23 | simpld 475 | . . . . . . . . . 10 |
25 | 3, 19 | fpwwelem 9467 | . . . . . . . . . 10 |
26 | 24, 25 | mpbid 222 | . . . . . . . . 9 |
27 | 26 | simprd 479 | . . . . . . . 8 |
28 | 27 | simpld 475 | . . . . . . 7 |
29 | fvex 6201 | . . . . . . . 8 | |
30 | weeq1 5102 | . . . . . . . 8 | |
31 | 29, 30 | spcev 3300 | . . . . . . 7 |
32 | 28, 31 | syl 17 | . . . . . 6 |
33 | ween 8858 | . . . . . 6 | |
34 | 32, 33 | sylibr 224 | . . . . 5 |
35 | 14, 34 | elind 3798 | . . . 4 |
36 | 8 | simp2d 1074 | . . . . . . . 8 |
37 | 36 | pssssd 3704 | . . . . . . 7 |
38 | 37, 11 | sstrd 3613 | . . . . . 6 |
39 | elpw2g 4827 | . . . . . . 7 | |
40 | 39 | adantr 481 | . . . . . 6 |
41 | 38, 40 | mpbird 247 | . . . . 5 |
42 | ssnum 8862 | . . . . . 6 | |
43 | 34, 37, 42 | syl2anc 693 | . . . . 5 |
44 | 41, 43 | elind 3798 | . . . 4 |
45 | f1fveq 6519 | . . . 4 | |
46 | 10, 35, 44, 45 | syl12anc 1324 | . . 3 |
47 | 9, 46 | mpbid 222 | . 2 |
48 | 36 | pssned 3705 | . . . 4 |
49 | 48 | necomd 2849 | . . 3 |
50 | 49 | neneqd 2799 | . 2 |
51 | 47, 50 | pm2.65da 600 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wex 1704 wcel 1990 wral 2912 cvv 3200 cin 3573 wss 3574 wpss 3575 cpw 4158 csn 4177 cuni 4436 class class class wbr 4653 copab 4712 wwe 5072 cxp 5112 ccnv 5113 cdm 5114 cima 5117 wf 5884 wf1 5885 cfv 5888 ccrd 8761 |
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-rep 4771 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-rmo 2920 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-1st 7168 df-wrecs 7407 df-recs 7468 df-er 7742 df-en 7956 df-dom 7957 df-oi 8415 df-card 8765 |
This theorem is referenced by: canthnum 9471 |
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