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Mirrors > Home > MPE Home > Th. List > dif1en | Structured version Visualization version Unicode version |
Description: If a set is equinumerous to the successor of a natural number , then with an element removed is equinumerous to . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) |
Ref | Expression |
---|---|
dif1en |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2 7086 | . . . . 5 | |
2 | breq2 4657 | . . . . . . 7 | |
3 | 2 | rspcev 3309 | . . . . . 6 |
4 | isfi 7979 | . . . . . 6 | |
5 | 3, 4 | sylibr 224 | . . . . 5 |
6 | 1, 5 | sylan 488 | . . . 4 |
7 | diffi 8192 | . . . . 5 | |
8 | isfi 7979 | . . . . 5 | |
9 | 7, 8 | sylib 208 | . . . 4 |
10 | 6, 9 | syl 17 | . . 3 |
11 | 10 | 3adant3 1081 | . 2 |
12 | vex 3203 | . . . . . . . 8 | |
13 | en2sn 8037 | . . . . . . . 8 | |
14 | 12, 13 | mpan2 707 | . . . . . . 7 |
15 | nnord 7073 | . . . . . . . 8 | |
16 | orddisj 5762 | . . . . . . . 8 | |
17 | 15, 16 | syl 17 | . . . . . . 7 |
18 | incom 3805 | . . . . . . . . . 10 | |
19 | disjdif 4040 | . . . . . . . . . 10 | |
20 | 18, 19 | eqtri 2644 | . . . . . . . . 9 |
21 | unen 8040 | . . . . . . . . . 10 | |
22 | 21 | an4s 869 | . . . . . . . . 9 |
23 | 20, 22 | mpanl2 717 | . . . . . . . 8 |
24 | 23 | expcom 451 | . . . . . . 7 |
25 | 14, 17, 24 | syl2an 494 | . . . . . 6 |
26 | 25 | 3ad2antl3 1225 | . . . . 5 |
27 | difsnid 4341 | . . . . . . . . 9 | |
28 | df-suc 5729 | . . . . . . . . . . 11 | |
29 | 28 | eqcomi 2631 | . . . . . . . . . 10 |
30 | 29 | a1i 11 | . . . . . . . . 9 |
31 | 27, 30 | breq12d 4666 | . . . . . . . 8 |
32 | 31 | 3ad2ant3 1084 | . . . . . . 7 |
33 | 32 | adantr 481 | . . . . . 6 |
34 | ensym 8005 | . . . . . . . . . . 11 | |
35 | entr 8008 | . . . . . . . . . . . . 13 | |
36 | peano2 7086 | . . . . . . . . . . . . . 14 | |
37 | nneneq 8143 | . . . . . . . . . . . . . 14 | |
38 | 36, 37 | sylan2 491 | . . . . . . . . . . . . 13 |
39 | 35, 38 | syl5ib 234 | . . . . . . . . . . . 12 |
40 | 39 | expd 452 | . . . . . . . . . . 11 |
41 | 34, 40 | syl5 34 | . . . . . . . . . 10 |
42 | 1, 41 | sylan 488 | . . . . . . . . 9 |
43 | 42 | imp 445 | . . . . . . . 8 |
44 | 43 | an32s 846 | . . . . . . 7 |
45 | 44 | 3adantl3 1219 | . . . . . 6 |
46 | 33, 45 | sylbid 230 | . . . . 5 |
47 | peano4 7088 | . . . . . . 7 | |
48 | 47 | biimpd 219 | . . . . . 6 |
49 | 48 | 3ad2antl1 1223 | . . . . 5 |
50 | 26, 46, 49 | 3syld 60 | . . . 4 |
51 | breq2 4657 | . . . . 5 | |
52 | 51 | biimprcd 240 | . . . 4 |
53 | 50, 52 | sylcom 30 | . . 3 |
54 | 53 | rexlimdva 3031 | . 2 |
55 | 11, 54 | mpd 15 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wrex 2913 cvv 3200 cdif 3571 cun 3572 cin 3573 c0 3915 csn 4177 class class class wbr 4653 word 5722 csuc 5725 com 7065 cen 7952 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-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: enp1i 8195 findcard 8199 findcard2 8200 en2eleq 8831 en2other2 8832 mreexexlem4d 16307 f1otrspeq 17867 pmtrf 17875 pmtrmvd 17876 pmtrfinv 17881 |
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