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Mirrors > Home > MPE Home > Th. List > enp1i | Structured version Visualization version Unicode version |
Description: Proof induction for en2i 7993 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
enp1i.1 | |
enp1i.2 | |
enp1i.3 | |
enp1i.4 |
Ref | Expression |
---|---|
enp1i |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsuceq0 5805 | . . . . 5 | |
2 | breq1 4656 | . . . . . . 7 | |
3 | enp1i.2 | . . . . . . . 8 | |
4 | ensym 8005 | . . . . . . . . 9 | |
5 | en0 8019 | . . . . . . . . 9 | |
6 | 4, 5 | sylib 208 | . . . . . . . 8 |
7 | 3, 6 | syl5eqr 2670 | . . . . . . 7 |
8 | 2, 7 | syl6bi 243 | . . . . . 6 |
9 | 8 | necon3ad 2807 | . . . . 5 |
10 | 1, 9 | mpi 20 | . . . 4 |
11 | 10 | con2i 134 | . . 3 |
12 | neq0 3930 | . . 3 | |
13 | 11, 12 | sylib 208 | . 2 |
14 | 3 | breq2i 4661 | . . . . 5 |
15 | enp1i.1 | . . . . . . . 8 | |
16 | dif1en 8193 | . . . . . . . 8 | |
17 | 15, 16 | mp3an1 1411 | . . . . . . 7 |
18 | enp1i.3 | . . . . . . 7 | |
19 | 17, 18 | syl 17 | . . . . . 6 |
20 | 19 | ex 450 | . . . . 5 |
21 | 14, 20 | sylbi 207 | . . . 4 |
22 | enp1i.4 | . . . 4 | |
23 | 21, 22 | sylcom 30 | . . 3 |
24 | 23 | eximdv 1846 | . 2 |
25 | 13, 24 | mpd 15 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wex 1704 wcel 1990 wne 2794 cdif 3571 c0 3915 csn 4177 class class class wbr 4653 csuc 5725 com 7065 cen 7952 |
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-rab 2921 df-v 3202 df-sbc 3436 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-br 4654 df-opab 4713 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-1o 7560 df-er 7742 df-en 7956 df-fin 7959 |
This theorem is referenced by: en2 8196 en3 8197 en4 8198 |
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