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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 993 | . . . 4 | |
2 | 1 | ensymd 6761 | . . 3 |
3 | bren 6725 | . . 3 | |
4 | 2, 3 | sylib 121 | . 2 |
5 | peano2 4579 | . . . . . . . 8 | |
6 | nnfi 6850 | . . . . . . . 8 | |
7 | 5, 6 | syl 14 | . . . . . . 7 |
8 | 7 | 3ad2ant1 1013 | . . . . . 6 |
9 | enfii 6852 | . . . . . 6 | |
10 | 8, 1, 9 | syl2anc 409 | . . . . 5 |
11 | 10 | adantr 274 | . . . 4 |
12 | simpl3 997 | . . . 4 | |
13 | f1of 5442 | . . . . . 6 | |
14 | 13 | adantl 275 | . . . . 5 |
15 | sucidg 4401 | . . . . . . 7 | |
16 | 15 | 3ad2ant1 1013 | . . . . . 6 |
17 | 16 | adantr 274 | . . . . 5 |
18 | 14, 17 | ffvelrnd 5632 | . . . 4 |
19 | fidifsnen 6848 | . . . 4 | |
20 | 11, 12, 18, 19 | syl3anc 1233 | . . 3 |
21 | nnord 4596 | . . . . . . . 8 | |
22 | orddif 4531 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | 23 | 3ad2ant1 1013 | . . . . . 6 |
25 | 24 | adantr 274 | . . . . 5 |
26 | 23 | eleq1d 2239 | . . . . . . . . 9 |
27 | 26 | ibi 175 | . . . . . . . 8 |
28 | 27 | 3ad2ant1 1013 | . . . . . . 7 |
29 | 28 | adantr 274 | . . . . . 6 |
30 | dff1o2 5447 | . . . . . . . . 9 | |
31 | 30 | simp2bi 1008 | . . . . . . . 8 |
32 | 31 | adantl 275 | . . . . . . 7 |
33 | f1ofo 5449 | . . . . . . . . 9 | |
34 | 33 | adantl 275 | . . . . . . . 8 |
35 | f1orel 5445 | . . . . . . . . . . . 12 | |
36 | 35 | adantl 275 | . . . . . . . . . . 11 |
37 | resdm 4930 | . . . . . . . . . . 11 | |
38 | 36, 37 | syl 14 | . . . . . . . . . 10 |
39 | f1odm 5446 | . . . . . . . . . . . 12 | |
40 | 39 | reseq2d 4891 | . . . . . . . . . . 11 |
41 | 40 | adantl 275 | . . . . . . . . . 10 |
42 | 38, 41 | eqtr3d 2205 | . . . . . . . . 9 |
43 | foeq1 5416 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 34, 44 | mpbid 146 | . . . . . . 7 |
46 | simpl1 995 | . . . . . . . . . 10 | |
47 | f1osng 5483 | . . . . . . . . . 10 | |
48 | 46, 18, 47 | syl2anc 409 | . . . . . . . . 9 |
49 | f1ofo 5449 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | f1ofn 5443 | . . . . . . . . . . 11 | |
52 | 51 | adantl 275 | . . . . . . . . . 10 |
53 | fnressn 5682 | . . . . . . . . . 10 | |
54 | 52, 17, 53 | syl2anc 409 | . . . . . . . . 9 |
55 | foeq1 5416 | . . . . . . . . 9 | |
56 | 54, 55 | syl 14 | . . . . . . . 8 |
57 | 50, 56 | mpbird 166 | . . . . . . 7 |
58 | resdif 5464 | . . . . . . 7 | |
59 | 32, 45, 57, 58 | syl3anc 1233 | . . . . . 6 |
60 | f1oeng 6735 | . . . . . 6 | |
61 | 29, 59, 60 | syl2anc 409 | . . . . 5 |
62 | 25, 61 | eqbrtrd 4011 | . . . 4 |
63 | 62 | ensymd 6761 | . . 3 |
64 | entr 6762 | . . 3 | |
65 | 20, 63, 64 | syl2anc 409 | . 2 |
66 | 4, 65 | exlimddv 1891 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wex 1485 wcel 2141 cdif 3118 csn 3583 cop 3586 class class class wbr 3989 word 4347 csuc 4350 com 4574 ccnv 4610 cdm 4611 crn 4612 cres 4613 wrel 4616 wfun 5192 wfn 5193 wf 5194 wfo 5196 wf1o 5197 cfv 5198 cen 6716 cfn 6718 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-er 6513 df-en 6719 df-fin 6721 |
This theorem is referenced by: dif1enen 6858 findcard 6866 findcard2 6867 findcard2s 6868 diffisn 6871 en2eleq 7172 en2other2 7173 zfz1isolem1 10775 |
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