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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 982 | . . . 4 | |
2 | 1 | ensymd 6677 | . . 3 |
3 | bren 6641 | . . 3 | |
4 | 2, 3 | sylib 121 | . 2 |
5 | peano2 4509 | . . . . . . . 8 | |
6 | nnfi 6766 | . . . . . . . 8 | |
7 | 5, 6 | syl 14 | . . . . . . 7 |
8 | 7 | 3ad2ant1 1002 | . . . . . 6 |
9 | enfii 6768 | . . . . . 6 | |
10 | 8, 1, 9 | syl2anc 408 | . . . . 5 |
11 | 10 | adantr 274 | . . . 4 |
12 | simpl3 986 | . . . 4 | |
13 | f1of 5367 | . . . . . 6 | |
14 | 13 | adantl 275 | . . . . 5 |
15 | sucidg 4338 | . . . . . . 7 | |
16 | 15 | 3ad2ant1 1002 | . . . . . 6 |
17 | 16 | adantr 274 | . . . . 5 |
18 | 14, 17 | ffvelrnd 5556 | . . . 4 |
19 | fidifsnen 6764 | . . . 4 | |
20 | 11, 12, 18, 19 | syl3anc 1216 | . . 3 |
21 | nnord 4525 | . . . . . . . 8 | |
22 | orddif 4462 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | 23 | 3ad2ant1 1002 | . . . . . 6 |
25 | 24 | adantr 274 | . . . . 5 |
26 | 23 | eleq1d 2208 | . . . . . . . . 9 |
27 | 26 | ibi 175 | . . . . . . . 8 |
28 | 27 | 3ad2ant1 1002 | . . . . . . 7 |
29 | 28 | adantr 274 | . . . . . 6 |
30 | dff1o2 5372 | . . . . . . . . 9 | |
31 | 30 | simp2bi 997 | . . . . . . . 8 |
32 | 31 | adantl 275 | . . . . . . 7 |
33 | f1ofo 5374 | . . . . . . . . 9 | |
34 | 33 | adantl 275 | . . . . . . . 8 |
35 | f1orel 5370 | . . . . . . . . . . . 12 | |
36 | 35 | adantl 275 | . . . . . . . . . . 11 |
37 | resdm 4858 | . . . . . . . . . . 11 | |
38 | 36, 37 | syl 14 | . . . . . . . . . 10 |
39 | f1odm 5371 | . . . . . . . . . . . 12 | |
40 | 39 | reseq2d 4819 | . . . . . . . . . . 11 |
41 | 40 | adantl 275 | . . . . . . . . . 10 |
42 | 38, 41 | eqtr3d 2174 | . . . . . . . . 9 |
43 | foeq1 5341 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 34, 44 | mpbid 146 | . . . . . . 7 |
46 | simpl1 984 | . . . . . . . . . 10 | |
47 | f1osng 5408 | . . . . . . . . . 10 | |
48 | 46, 18, 47 | syl2anc 408 | . . . . . . . . 9 |
49 | f1ofo 5374 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | f1ofn 5368 | . . . . . . . . . . 11 | |
52 | 51 | adantl 275 | . . . . . . . . . 10 |
53 | fnressn 5606 | . . . . . . . . . 10 | |
54 | 52, 17, 53 | syl2anc 408 | . . . . . . . . 9 |
55 | foeq1 5341 | . . . . . . . . 9 | |
56 | 54, 55 | syl 14 | . . . . . . . 8 |
57 | 50, 56 | mpbird 166 | . . . . . . 7 |
58 | resdif 5389 | . . . . . . 7 | |
59 | 32, 45, 57, 58 | syl3anc 1216 | . . . . . 6 |
60 | f1oeng 6651 | . . . . . 6 | |
61 | 29, 59, 60 | syl2anc 408 | . . . . 5 |
62 | 25, 61 | eqbrtrd 3950 | . . . 4 |
63 | 62 | ensymd 6677 | . . 3 |
64 | entr 6678 | . . 3 | |
65 | 20, 63, 64 | syl2anc 408 | . 2 |
66 | 4, 65 | exlimddv 1870 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wex 1468 wcel 1480 cdif 3068 csn 3527 cop 3530 class class class wbr 3929 word 4284 csuc 4287 com 4504 ccnv 4538 cdm 4539 crn 4540 cres 4541 wrel 4544 wfun 5117 wfn 5118 wf 5119 wfo 5121 wf1o 5122 cfv 5123 cen 6632 cfn 6634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: dif1enen 6774 findcard 6782 findcard2 6783 findcard2s 6784 diffisn 6787 en2eleq 7051 en2other2 7052 zfz1isolem1 10583 |
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