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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 983 | . . . 4 | |
2 | 1 | ensymd 6729 | . . 3 |
3 | bren 6693 | . . 3 | |
4 | 2, 3 | sylib 121 | . 2 |
5 | peano2 4555 | . . . . . . . 8 | |
6 | nnfi 6818 | . . . . . . . 8 | |
7 | 5, 6 | syl 14 | . . . . . . 7 |
8 | 7 | 3ad2ant1 1003 | . . . . . 6 |
9 | enfii 6820 | . . . . . 6 | |
10 | 8, 1, 9 | syl2anc 409 | . . . . 5 |
11 | 10 | adantr 274 | . . . 4 |
12 | simpl3 987 | . . . 4 | |
13 | f1of 5415 | . . . . . 6 | |
14 | 13 | adantl 275 | . . . . 5 |
15 | sucidg 4377 | . . . . . . 7 | |
16 | 15 | 3ad2ant1 1003 | . . . . . 6 |
17 | 16 | adantr 274 | . . . . 5 |
18 | 14, 17 | ffvelrnd 5604 | . . . 4 |
19 | fidifsnen 6816 | . . . 4 | |
20 | 11, 12, 18, 19 | syl3anc 1220 | . . 3 |
21 | nnord 4572 | . . . . . . . 8 | |
22 | orddif 4507 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | 23 | 3ad2ant1 1003 | . . . . . 6 |
25 | 24 | adantr 274 | . . . . 5 |
26 | 23 | eleq1d 2226 | . . . . . . . . 9 |
27 | 26 | ibi 175 | . . . . . . . 8 |
28 | 27 | 3ad2ant1 1003 | . . . . . . 7 |
29 | 28 | adantr 274 | . . . . . 6 |
30 | dff1o2 5420 | . . . . . . . . 9 | |
31 | 30 | simp2bi 998 | . . . . . . . 8 |
32 | 31 | adantl 275 | . . . . . . 7 |
33 | f1ofo 5422 | . . . . . . . . 9 | |
34 | 33 | adantl 275 | . . . . . . . 8 |
35 | f1orel 5418 | . . . . . . . . . . . 12 | |
36 | 35 | adantl 275 | . . . . . . . . . . 11 |
37 | resdm 4906 | . . . . . . . . . . 11 | |
38 | 36, 37 | syl 14 | . . . . . . . . . 10 |
39 | f1odm 5419 | . . . . . . . . . . . 12 | |
40 | 39 | reseq2d 4867 | . . . . . . . . . . 11 |
41 | 40 | adantl 275 | . . . . . . . . . 10 |
42 | 38, 41 | eqtr3d 2192 | . . . . . . . . 9 |
43 | foeq1 5389 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 34, 44 | mpbid 146 | . . . . . . 7 |
46 | simpl1 985 | . . . . . . . . . 10 | |
47 | f1osng 5456 | . . . . . . . . . 10 | |
48 | 46, 18, 47 | syl2anc 409 | . . . . . . . . 9 |
49 | f1ofo 5422 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | f1ofn 5416 | . . . . . . . . . . 11 | |
52 | 51 | adantl 275 | . . . . . . . . . 10 |
53 | fnressn 5654 | . . . . . . . . . 10 | |
54 | 52, 17, 53 | syl2anc 409 | . . . . . . . . 9 |
55 | foeq1 5389 | . . . . . . . . 9 | |
56 | 54, 55 | syl 14 | . . . . . . . 8 |
57 | 50, 56 | mpbird 166 | . . . . . . 7 |
58 | resdif 5437 | . . . . . . 7 | |
59 | 32, 45, 57, 58 | syl3anc 1220 | . . . . . 6 |
60 | f1oeng 6703 | . . . . . 6 | |
61 | 29, 59, 60 | syl2anc 409 | . . . . 5 |
62 | 25, 61 | eqbrtrd 3987 | . . . 4 |
63 | 62 | ensymd 6729 | . . 3 |
64 | entr 6730 | . . 3 | |
65 | 20, 63, 64 | syl2anc 409 | . 2 |
66 | 4, 65 | exlimddv 1878 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wex 1472 wcel 2128 cdif 3099 csn 3560 cop 3563 class class class wbr 3966 word 4323 csuc 4326 com 4550 ccnv 4586 cdm 4587 crn 4588 cres 4589 wrel 4592 wfun 5165 wfn 5166 wf 5167 wfo 5169 wf1o 5170 cfv 5171 cen 6684 cfn 6686 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-er 6481 df-en 6687 df-fin 6689 |
This theorem is referenced by: dif1enen 6826 findcard 6834 findcard2 6835 findcard2s 6836 diffisn 6839 en2eleq 7131 en2other2 7132 zfz1isolem1 10715 |
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