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Mirrors > Home > MPE Home > Th. List > fineqvlem | Structured version Visualization version Unicode version |
Description: Lemma for fineqv 8175. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
fineqvlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4850 | . . . 4 | |
2 | 1 | adantr 481 | . . 3 |
3 | pwexg 4850 | . . 3 | |
4 | 2, 3 | syl 17 | . 2 |
5 | ssrab2 3687 | . . . . 5 | |
6 | elpw2g 4827 | . . . . . 6 | |
7 | 2, 6 | syl 17 | . . . . 5 |
8 | 5, 7 | mpbiri 248 | . . . 4 |
9 | 8 | a1d 25 | . . 3 |
10 | isinf 8173 | . . . . . . . . 9 | |
11 | 10 | r19.21bi 2932 | . . . . . . . 8 |
12 | 11 | ad2ant2lr 784 | . . . . . . 7 |
13 | selpw 4165 | . . . . . . . . . . 11 | |
14 | 13 | biimpri 218 | . . . . . . . . . 10 |
15 | 14 | anim1i 592 | . . . . . . . . 9 |
16 | breq1 4656 | . . . . . . . . . 10 | |
17 | 16 | elrab 3363 | . . . . . . . . 9 |
18 | 15, 17 | sylibr 224 | . . . . . . . 8 |
19 | 18 | eximi 1762 | . . . . . . 7 |
20 | 12, 19 | syl 17 | . . . . . 6 |
21 | eleq2 2690 | . . . . . . . . 9 | |
22 | 21 | biimpcd 239 | . . . . . . . 8 |
23 | 22 | adantl 482 | . . . . . . 7 |
24 | 17 | simprbi 480 | . . . . . . . . . 10 |
25 | breq1 4656 | . . . . . . . . . . . 12 | |
26 | 25 | elrab 3363 | . . . . . . . . . . 11 |
27 | 26 | simprbi 480 | . . . . . . . . . 10 |
28 | ensym 8005 | . . . . . . . . . . 11 | |
29 | entr 8008 | . . . . . . . . . . 11 | |
30 | 28, 29 | sylan 488 | . . . . . . . . . 10 |
31 | 24, 27, 30 | syl2an 494 | . . . . . . . . 9 |
32 | 31 | ex 450 | . . . . . . . 8 |
33 | 32 | adantl 482 | . . . . . . 7 |
34 | nneneq 8143 | . . . . . . . . 9 | |
35 | 34 | biimpd 219 | . . . . . . . 8 |
36 | 35 | ad2antlr 763 | . . . . . . 7 |
37 | 23, 33, 36 | 3syld 60 | . . . . . 6 |
38 | 20, 37 | exlimddv 1863 | . . . . 5 |
39 | breq2 4657 | . . . . . 6 | |
40 | 39 | rabbidv 3189 | . . . . 5 |
41 | 38, 40 | impbid1 215 | . . . 4 |
42 | 41 | ex 450 | . . 3 |
43 | 9, 42 | dom2d 7996 | . 2 |
44 | 4, 43 | mpd 15 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 crab 2916 cvv 3200 wss 3574 cpw 4158 class class class wbr 4653 com 7065 cen 7952 cdom 7953 cfn 7955 |
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-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-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-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-er 7742 df-en 7956 df-dom 7957 df-fin 7959 |
This theorem is referenced by: fineqv 8175 isfin1-2 9207 |
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