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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 967 | . . . 4 | |
2 | 1 | ensymd 6645 | . . 3 |
3 | bren 6609 | . . 3 | |
4 | 2, 3 | sylib 121 | . 2 |
5 | peano2 4479 | . . . . . . . 8 | |
6 | nnfi 6734 | . . . . . . . 8 | |
7 | 5, 6 | syl 14 | . . . . . . 7 |
8 | 7 | 3ad2ant1 987 | . . . . . 6 |
9 | enfii 6736 | . . . . . 6 | |
10 | 8, 1, 9 | syl2anc 408 | . . . . 5 |
11 | 10 | adantr 274 | . . . 4 |
12 | simpl3 971 | . . . 4 | |
13 | f1of 5335 | . . . . . 6 | |
14 | 13 | adantl 275 | . . . . 5 |
15 | sucidg 4308 | . . . . . . 7 | |
16 | 15 | 3ad2ant1 987 | . . . . . 6 |
17 | 16 | adantr 274 | . . . . 5 |
18 | 14, 17 | ffvelrnd 5524 | . . . 4 |
19 | fidifsnen 6732 | . . . 4 | |
20 | 11, 12, 18, 19 | syl3anc 1201 | . . 3 |
21 | nnord 4495 | . . . . . . . 8 | |
22 | orddif 4432 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | 23 | 3ad2ant1 987 | . . . . . 6 |
25 | 24 | adantr 274 | . . . . 5 |
26 | 23 | eleq1d 2186 | . . . . . . . . 9 |
27 | 26 | ibi 175 | . . . . . . . 8 |
28 | 27 | 3ad2ant1 987 | . . . . . . 7 |
29 | 28 | adantr 274 | . . . . . 6 |
30 | dff1o2 5340 | . . . . . . . . 9 | |
31 | 30 | simp2bi 982 | . . . . . . . 8 |
32 | 31 | adantl 275 | . . . . . . 7 |
33 | f1ofo 5342 | . . . . . . . . 9 | |
34 | 33 | adantl 275 | . . . . . . . 8 |
35 | f1orel 5338 | . . . . . . . . . . . 12 | |
36 | 35 | adantl 275 | . . . . . . . . . . 11 |
37 | resdm 4828 | . . . . . . . . . . 11 | |
38 | 36, 37 | syl 14 | . . . . . . . . . 10 |
39 | f1odm 5339 | . . . . . . . . . . . 12 | |
40 | 39 | reseq2d 4789 | . . . . . . . . . . 11 |
41 | 40 | adantl 275 | . . . . . . . . . 10 |
42 | 38, 41 | eqtr3d 2152 | . . . . . . . . 9 |
43 | foeq1 5311 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 34, 44 | mpbid 146 | . . . . . . 7 |
46 | simpl1 969 | . . . . . . . . . 10 | |
47 | f1osng 5376 | . . . . . . . . . 10 | |
48 | 46, 18, 47 | syl2anc 408 | . . . . . . . . 9 |
49 | f1ofo 5342 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | f1ofn 5336 | . . . . . . . . . . 11 | |
52 | 51 | adantl 275 | . . . . . . . . . 10 |
53 | fnressn 5574 | . . . . . . . . . 10 | |
54 | 52, 17, 53 | syl2anc 408 | . . . . . . . . 9 |
55 | foeq1 5311 | . . . . . . . . 9 | |
56 | 54, 55 | syl 14 | . . . . . . . 8 |
57 | 50, 56 | mpbird 166 | . . . . . . 7 |
58 | resdif 5357 | . . . . . . 7 | |
59 | 32, 45, 57, 58 | syl3anc 1201 | . . . . . 6 |
60 | f1oeng 6619 | . . . . . 6 | |
61 | 29, 59, 60 | syl2anc 408 | . . . . 5 |
62 | 25, 61 | eqbrtrd 3920 | . . . 4 |
63 | 62 | ensymd 6645 | . . 3 |
64 | entr 6646 | . . 3 | |
65 | 20, 63, 64 | syl2anc 408 | . 2 |
66 | 4, 65 | exlimddv 1854 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wex 1453 wcel 1465 cdif 3038 csn 3497 cop 3500 class class class wbr 3899 word 4254 csuc 4257 com 4474 ccnv 4508 cdm 4509 crn 4510 cres 4511 wrel 4514 wfun 5087 wfn 5088 wf 5089 wfo 5091 wf1o 5092 cfv 5093 cen 6600 cfn 6602 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: dif1enen 6742 findcard 6750 findcard2 6751 findcard2s 6752 diffisn 6755 en2eleq 7019 en2other2 7020 zfz1isolem1 10551 |
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