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Mirrors > Home > ILE Home > Th. List > dif1en | 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 | simp2 905 | . . . 4 | |
2 | 1 | ensymd 6263 | . . 3 |
3 | bren 6228 | . . 3 | |
4 | 2, 3 | sylib 127 | . 2 |
5 | peano2 4318 | . . . . . . . 8 | |
6 | nnfi 6333 | . . . . . . . 8 | |
7 | 5, 6 | syl 14 | . . . . . . 7 |
8 | 7 | 3ad2ant1 925 | . . . . . 6 |
9 | enfii 6335 | . . . . . 6 | |
10 | 8, 1, 9 | syl2anc 391 | . . . . 5 |
11 | 10 | adantr 261 | . . . 4 |
12 | simpl3 909 | . . . 4 | |
13 | f1of 5126 | . . . . . 6 | |
14 | 13 | adantl 262 | . . . . 5 |
15 | sucidg 4153 | . . . . . . 7 | |
16 | 15 | 3ad2ant1 925 | . . . . . 6 |
17 | 16 | adantr 261 | . . . . 5 |
18 | 14, 17 | ffvelrnd 5303 | . . . 4 |
19 | fidifsnen 6331 | . . . 4 | |
20 | 11, 12, 18, 19 | syl3anc 1135 | . . 3 |
21 | nnord 4334 | . . . . . . . 8 | |
22 | orddif 4271 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | 23 | 3ad2ant1 925 | . . . . . 6 |
25 | 24 | adantr 261 | . . . . 5 |
26 | 23 | eleq1d 2106 | . . . . . . . . 9 |
27 | 26 | ibi 165 | . . . . . . . 8 |
28 | 27 | 3ad2ant1 925 | . . . . . . 7 |
29 | 28 | adantr 261 | . . . . . 6 |
30 | dff1o2 5131 | . . . . . . . . 9 | |
31 | 30 | simp2bi 920 | . . . . . . . 8 |
32 | 31 | adantl 262 | . . . . . . 7 |
33 | f1ofo 5133 | . . . . . . . . 9 | |
34 | 33 | adantl 262 | . . . . . . . 8 |
35 | f1orel 5129 | . . . . . . . . . . . 12 | |
36 | 35 | adantl 262 | . . . . . . . . . . 11 |
37 | resdm 4649 | . . . . . . . . . . 11 | |
38 | 36, 37 | syl 14 | . . . . . . . . . 10 |
39 | f1odm 5130 | . . . . . . . . . . . 12 | |
40 | 39 | reseq2d 4612 | . . . . . . . . . . 11 |
41 | 40 | adantl 262 | . . . . . . . . . 10 |
42 | 38, 41 | eqtr3d 2074 | . . . . . . . . 9 |
43 | foeq1 5102 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 34, 44 | mpbid 135 | . . . . . . 7 |
46 | simpl1 907 | . . . . . . . . . 10 | |
47 | f1osng 5167 | . . . . . . . . . 10 | |
48 | 46, 18, 47 | syl2anc 391 | . . . . . . . . 9 |
49 | f1ofo 5133 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | f1ofn 5127 | . . . . . . . . . . 11 | |
52 | 51 | adantl 262 | . . . . . . . . . 10 |
53 | fnressn 5349 | . . . . . . . . . 10 | |
54 | 52, 17, 53 | syl2anc 391 | . . . . . . . . 9 |
55 | foeq1 5102 | . . . . . . . . 9 | |
56 | 54, 55 | syl 14 | . . . . . . . 8 |
57 | 50, 56 | mpbird 156 | . . . . . . 7 |
58 | resdif 5148 | . . . . . . 7 | |
59 | 32, 45, 57, 58 | syl3anc 1135 | . . . . . 6 |
60 | f1oeng 6237 | . . . . . 6 | |
61 | 29, 59, 60 | syl2anc 391 | . . . . 5 |
62 | 25, 61 | eqbrtrd 3784 | . . . 4 |
63 | 62 | ensymd 6263 | . . 3 |
64 | entr 6264 | . . 3 | |
65 | 20, 63, 64 | syl2anc 391 | . 2 |
66 | 4, 65 | exlimddv 1778 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wex 1381 wcel 1393 cdif 2914 csn 3375 cop 3378 class class class wbr 3764 word 4099 csuc 4102 com 4313 ccnv 4344 cdm 4345 crn 4346 cres 4347 wrel 4350 wfun 4896 wfn 4897 wf 4898 wfo 4900 wf1o 4901 cfv 4902 cen 6219 cfn 6221 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-er 6106 df-en 6222 df-fin 6224 |
This theorem is referenced by: findcard 6345 findcard2 6346 findcard2s 6347 diffisn 6350 |
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