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Mirrors > Home > ILE Home > Th. List > fidifsnen | Unicode version |
Description: All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.) |
Ref | Expression |
---|---|
fidifsnen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexg 4069 | . . . . . 6 | |
2 | 1 | 3ad2ant1 1002 | . . . . 5 |
3 | 2 | adantr 274 | . . . 4 |
4 | enrefg 6658 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | sneq 3538 | . . . . 5 | |
7 | 6 | difeq2d 3194 | . . . 4 |
8 | 7 | adantl 275 | . . 3 |
9 | 5, 8 | breqtrd 3954 | . 2 |
10 | 2 | adantr 274 | . . 3 |
11 | eqid 2139 | . . . 4 | |
12 | iftrue 3479 | . . . . . . . 8 | |
13 | 12 | adantl 275 | . . . . . . 7 |
14 | simpll2 1021 | . . . . . . . 8 | |
15 | 14 | adantr 274 | . . . . . . 7 |
16 | 13, 15 | eqeltrd 2216 | . . . . . 6 |
17 | simpllr 523 | . . . . . . . 8 | |
18 | 13 | eqeq1d 2148 | . . . . . . . 8 |
19 | 17, 18 | mtbird 662 | . . . . . . 7 |
20 | 19 | neneqad 2387 | . . . . . 6 |
21 | eldifsn 3650 | . . . . . 6 | |
22 | 16, 20, 21 | sylanbrc 413 | . . . . 5 |
23 | iffalse 3482 | . . . . . . . 8 | |
24 | 23 | adantl 275 | . . . . . . 7 |
25 | eldifi 3198 | . . . . . . . 8 | |
26 | 25 | ad2antlr 480 | . . . . . . 7 |
27 | 24, 26 | eqeltrd 2216 | . . . . . 6 |
28 | simpr 109 | . . . . . . . 8 | |
29 | 24 | eqeq1d 2148 | . . . . . . . 8 |
30 | 28, 29 | mtbird 662 | . . . . . . 7 |
31 | 30 | neneqad 2387 | . . . . . 6 |
32 | 27, 31, 21 | sylanbrc 413 | . . . . 5 |
33 | simpll1 1020 | . . . . . . 7 | |
34 | 25 | adantl 275 | . . . . . . 7 |
35 | simpll3 1022 | . . . . . . 7 | |
36 | fidceq 6763 | . . . . . . 7 DECID | |
37 | 33, 34, 35, 36 | syl3anc 1216 | . . . . . 6 DECID |
38 | exmiddc 821 | . . . . . 6 DECID | |
39 | 37, 38 | syl 14 | . . . . 5 |
40 | 22, 32, 39 | mpjaodan 787 | . . . 4 |
41 | iftrue 3479 | . . . . . . 7 | |
42 | 41 | adantl 275 | . . . . . 6 |
43 | simpl3 986 | . . . . . . . 8 | |
44 | simpr 109 | . . . . . . . . . 10 | |
45 | 44 | neneqad 2387 | . . . . . . . . 9 |
46 | 45 | necomd 2394 | . . . . . . . 8 |
47 | eldifsn 3650 | . . . . . . . 8 | |
48 | 43, 46, 47 | sylanbrc 413 | . . . . . . 7 |
49 | 48 | ad2antrr 479 | . . . . . 6 |
50 | 42, 49 | eqeltrd 2216 | . . . . 5 |
51 | iffalse 3482 | . . . . . . 7 | |
52 | 51 | adantl 275 | . . . . . 6 |
53 | eldifi 3198 | . . . . . . . 8 | |
54 | 53 | ad2antlr 480 | . . . . . . 7 |
55 | simpr 109 | . . . . . . . 8 | |
56 | 55 | neneqad 2387 | . . . . . . 7 |
57 | eldifsn 3650 | . . . . . . 7 | |
58 | 54, 56, 57 | sylanbrc 413 | . . . . . 6 |
59 | 52, 58 | eqeltrd 2216 | . . . . 5 |
60 | simpll1 1020 | . . . . . . 7 | |
61 | 53 | adantl 275 | . . . . . . 7 |
62 | simpll2 1021 | . . . . . . 7 | |
63 | fidceq 6763 | . . . . . . 7 DECID | |
64 | 60, 61, 62, 63 | syl3anc 1216 | . . . . . 6 DECID |
65 | exmiddc 821 | . . . . . 6 DECID | |
66 | 64, 65 | syl 14 | . . . . 5 |
67 | 50, 59, 66 | mpjaodan 787 | . . . 4 |
68 | 12 | adantl 275 | . . . . . . . . . 10 |
69 | 68 | eqeq2d 2151 | . . . . . . . . 9 |
70 | 69 | biimpar 295 | . . . . . . . 8 |
71 | 70 | a1d 22 | . . . . . . 7 |
72 | simpr 109 | . . . . . . . . . . 11 | |
73 | 51 | eqeq2d 2151 | . . . . . . . . . . . 12 |
74 | 73 | ad2antlr 480 | . . . . . . . . . . 11 |
75 | 72, 74 | mpbid 146 | . . . . . . . . . 10 |
76 | simpllr 523 | . . . . . . . . . 10 | |
77 | 75, 76 | eqtr3d 2174 | . . . . . . . . 9 |
78 | simprr 521 | . . . . . . . . . . . . 13 | |
79 | 78 | ad2antrr 479 | . . . . . . . . . . . 12 |
80 | 79 | eldifbd 3083 | . . . . . . . . . . 11 |
81 | 80 | adantr 274 | . . . . . . . . . 10 |
82 | velsn 3544 | . . . . . . . . . 10 | |
83 | 81, 82 | sylnib 665 | . . . . . . . . 9 |
84 | 77, 83 | pm2.21dd 609 | . . . . . . . 8 |
85 | 84 | ex 114 | . . . . . . 7 |
86 | simpll1 1020 | . . . . . . . . . 10 | |
87 | 53 | ad2antll 482 | . . . . . . . . . 10 |
88 | simpll2 1021 | . . . . . . . . . 10 | |
89 | 86, 87, 88, 63 | syl3anc 1216 | . . . . . . . . 9 DECID |
90 | 89, 65 | syl 14 | . . . . . . . 8 |
91 | 90 | adantr 274 | . . . . . . 7 |
92 | 71, 85, 91 | mpjaodan 787 | . . . . . 6 |
93 | 41 | eqeq2d 2151 | . . . . . . . . 9 |
94 | 93 | biimprcd 159 | . . . . . . . 8 |
95 | 94 | adantl 275 | . . . . . . 7 |
96 | 69, 95 | sylbid 149 | . . . . . 6 |
97 | 92, 96 | impbid 128 | . . . . 5 |
98 | simplr 519 | . . . . . . . . 9 | |
99 | 41 | adantl 275 | . . . . . . . . 9 |
100 | 98, 99 | eqtrd 2172 | . . . . . . . 8 |
101 | simpllr 523 | . . . . . . . 8 | |
102 | 100, 101 | pm2.21dd 609 | . . . . . . 7 |
103 | 23 | ad3antlr 484 | . . . . . . . 8 |
104 | simplr 519 | . . . . . . . . 9 | |
105 | 51 | adantl 275 | . . . . . . . . 9 |
106 | 104, 105 | eqtrd 2172 | . . . . . . . 8 |
107 | 103, 106 | eqtr2d 2173 | . . . . . . 7 |
108 | 90 | ad2antrr 479 | . . . . . . 7 |
109 | 102, 107, 108 | mpjaodan 787 | . . . . . 6 |
110 | simprl 520 | . . . . . . . . . . . 12 | |
111 | 110 | eldifbd 3083 | . . . . . . . . . . 11 |
112 | velsn 3544 | . . . . . . . . . . 11 | |
113 | 111, 112 | sylnib 665 | . . . . . . . . . 10 |
114 | 113 | ad2antrr 479 | . . . . . . . . 9 |
115 | simpr 109 | . . . . . . . . . . 11 | |
116 | 23 | eqeq2d 2151 | . . . . . . . . . . . 12 |
117 | 116 | ad2antlr 480 | . . . . . . . . . . 11 |
118 | 115, 117 | mpbid 146 | . . . . . . . . . 10 |
119 | 118 | eqeq1d 2148 | . . . . . . . . 9 |
120 | 114, 119 | mtbird 662 | . . . . . . . 8 |
121 | 120, 51 | syl 14 | . . . . . . 7 |
122 | 121, 118 | eqtr2d 2173 | . . . . . 6 |
123 | 109, 122 | impbida 585 | . . . . 5 |
124 | 39 | adantrr 470 | . . . . 5 |
125 | 97, 123, 124 | mpjaodan 787 | . . . 4 |
126 | 11, 40, 67, 125 | f1o2d 5975 | . . 3 |
127 | f1oeng 6651 | . . 3 | |
128 | 10, 126, 127 | syl2anc 408 | . 2 |
129 | fidceq 6763 | . . 3 DECID | |
130 | exmiddc 821 | . . 3 DECID | |
131 | 129, 130 | syl 14 | . 2 |
132 | 9, 128, 131 | mpjaodan 787 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 w3a 962 wceq 1331 wcel 1480 wne 2308 cvv 2686 cdif 3068 cif 3474 csn 3527 class class class wbr 3929 cmpt 3989 wf1o 5122 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-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-en 6635 df-fin 6637 |
This theorem is referenced by: dif1en 6773 |
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