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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 4039 | . . . . . 6 | |
2 | 1 | 3ad2ant1 987 | . . . . 5 |
3 | 2 | adantr 274 | . . . 4 |
4 | enrefg 6626 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | sneq 3508 | . . . . 5 | |
7 | 6 | difeq2d 3164 | . . . 4 |
8 | 7 | adantl 275 | . . 3 |
9 | 5, 8 | breqtrd 3924 | . 2 |
10 | 2 | adantr 274 | . . 3 |
11 | eqid 2117 | . . . 4 | |
12 | iftrue 3449 | . . . . . . . 8 | |
13 | 12 | adantl 275 | . . . . . . 7 |
14 | simpll2 1006 | . . . . . . . 8 | |
15 | 14 | adantr 274 | . . . . . . 7 |
16 | 13, 15 | eqeltrd 2194 | . . . . . 6 |
17 | simpllr 508 | . . . . . . . 8 | |
18 | 13 | eqeq1d 2126 | . . . . . . . 8 |
19 | 17, 18 | mtbird 647 | . . . . . . 7 |
20 | 19 | neneqad 2364 | . . . . . 6 |
21 | eldifsn 3620 | . . . . . 6 | |
22 | 16, 20, 21 | sylanbrc 413 | . . . . 5 |
23 | iffalse 3452 | . . . . . . . 8 | |
24 | 23 | adantl 275 | . . . . . . 7 |
25 | eldifi 3168 | . . . . . . . 8 | |
26 | 25 | ad2antlr 480 | . . . . . . 7 |
27 | 24, 26 | eqeltrd 2194 | . . . . . 6 |
28 | simpr 109 | . . . . . . . 8 | |
29 | 24 | eqeq1d 2126 | . . . . . . . 8 |
30 | 28, 29 | mtbird 647 | . . . . . . 7 |
31 | 30 | neneqad 2364 | . . . . . 6 |
32 | 27, 31, 21 | sylanbrc 413 | . . . . 5 |
33 | simpll1 1005 | . . . . . . 7 | |
34 | 25 | adantl 275 | . . . . . . 7 |
35 | simpll3 1007 | . . . . . . 7 | |
36 | fidceq 6731 | . . . . . . 7 DECID | |
37 | 33, 34, 35, 36 | syl3anc 1201 | . . . . . 6 DECID |
38 | exmiddc 806 | . . . . . 6 DECID | |
39 | 37, 38 | syl 14 | . . . . 5 |
40 | 22, 32, 39 | mpjaodan 772 | . . . 4 |
41 | iftrue 3449 | . . . . . . 7 | |
42 | 41 | adantl 275 | . . . . . 6 |
43 | simpl3 971 | . . . . . . . 8 | |
44 | simpr 109 | . . . . . . . . . 10 | |
45 | 44 | neneqad 2364 | . . . . . . . . 9 |
46 | 45 | necomd 2371 | . . . . . . . 8 |
47 | eldifsn 3620 | . . . . . . . 8 | |
48 | 43, 46, 47 | sylanbrc 413 | . . . . . . 7 |
49 | 48 | ad2antrr 479 | . . . . . 6 |
50 | 42, 49 | eqeltrd 2194 | . . . . 5 |
51 | iffalse 3452 | . . . . . . 7 | |
52 | 51 | adantl 275 | . . . . . 6 |
53 | eldifi 3168 | . . . . . . . 8 | |
54 | 53 | ad2antlr 480 | . . . . . . 7 |
55 | simpr 109 | . . . . . . . 8 | |
56 | 55 | neneqad 2364 | . . . . . . 7 |
57 | eldifsn 3620 | . . . . . . 7 | |
58 | 54, 56, 57 | sylanbrc 413 | . . . . . 6 |
59 | 52, 58 | eqeltrd 2194 | . . . . 5 |
60 | simpll1 1005 | . . . . . . 7 | |
61 | 53 | adantl 275 | . . . . . . 7 |
62 | simpll2 1006 | . . . . . . 7 | |
63 | fidceq 6731 | . . . . . . 7 DECID | |
64 | 60, 61, 62, 63 | syl3anc 1201 | . . . . . 6 DECID |
65 | exmiddc 806 | . . . . . 6 DECID | |
66 | 64, 65 | syl 14 | . . . . 5 |
67 | 50, 59, 66 | mpjaodan 772 | . . . 4 |
68 | 12 | adantl 275 | . . . . . . . . . 10 |
69 | 68 | eqeq2d 2129 | . . . . . . . . 9 |
70 | 69 | biimpar 295 | . . . . . . . 8 |
71 | 70 | a1d 22 | . . . . . . 7 |
72 | simpr 109 | . . . . . . . . . . 11 | |
73 | 51 | eqeq2d 2129 | . . . . . . . . . . . 12 |
74 | 73 | ad2antlr 480 | . . . . . . . . . . 11 |
75 | 72, 74 | mpbid 146 | . . . . . . . . . 10 |
76 | simpllr 508 | . . . . . . . . . 10 | |
77 | 75, 76 | eqtr3d 2152 | . . . . . . . . 9 |
78 | simprr 506 | . . . . . . . . . . . . 13 | |
79 | 78 | ad2antrr 479 | . . . . . . . . . . . 12 |
80 | 79 | eldifbd 3053 | . . . . . . . . . . 11 |
81 | 80 | adantr 274 | . . . . . . . . . 10 |
82 | velsn 3514 | . . . . . . . . . 10 | |
83 | 81, 82 | sylnib 650 | . . . . . . . . 9 |
84 | 77, 83 | pm2.21dd 594 | . . . . . . . 8 |
85 | 84 | ex 114 | . . . . . . 7 |
86 | simpll1 1005 | . . . . . . . . . 10 | |
87 | 53 | ad2antll 482 | . . . . . . . . . 10 |
88 | simpll2 1006 | . . . . . . . . . 10 | |
89 | 86, 87, 88, 63 | syl3anc 1201 | . . . . . . . . 9 DECID |
90 | 89, 65 | syl 14 | . . . . . . . 8 |
91 | 90 | adantr 274 | . . . . . . 7 |
92 | 71, 85, 91 | mpjaodan 772 | . . . . . 6 |
93 | 41 | eqeq2d 2129 | . . . . . . . . 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 504 | . . . . . . . . 9 | |
99 | 41 | adantl 275 | . . . . . . . . 9 |
100 | 98, 99 | eqtrd 2150 | . . . . . . . 8 |
101 | simpllr 508 | . . . . . . . 8 | |
102 | 100, 101 | pm2.21dd 594 | . . . . . . 7 |
103 | 23 | ad3antlr 484 | . . . . . . . 8 |
104 | simplr 504 | . . . . . . . . 9 | |
105 | 51 | adantl 275 | . . . . . . . . 9 |
106 | 104, 105 | eqtrd 2150 | . . . . . . . 8 |
107 | 103, 106 | eqtr2d 2151 | . . . . . . 7 |
108 | 90 | ad2antrr 479 | . . . . . . 7 |
109 | 102, 107, 108 | mpjaodan 772 | . . . . . 6 |
110 | simprl 505 | . . . . . . . . . . . 12 | |
111 | 110 | eldifbd 3053 | . . . . . . . . . . 11 |
112 | velsn 3514 | . . . . . . . . . . 11 | |
113 | 111, 112 | sylnib 650 | . . . . . . . . . 10 |
114 | 113 | ad2antrr 479 | . . . . . . . . 9 |
115 | simpr 109 | . . . . . . . . . . 11 | |
116 | 23 | eqeq2d 2129 | . . . . . . . . . . . 12 |
117 | 116 | ad2antlr 480 | . . . . . . . . . . 11 |
118 | 115, 117 | mpbid 146 | . . . . . . . . . 10 |
119 | 118 | eqeq1d 2126 | . . . . . . . . 9 |
120 | 114, 119 | mtbird 647 | . . . . . . . 8 |
121 | 120, 51 | syl 14 | . . . . . . 7 |
122 | 121, 118 | eqtr2d 2151 | . . . . . 6 |
123 | 109, 122 | impbida 570 | . . . . 5 |
124 | 39 | adantrr 470 | . . . . 5 |
125 | 97, 123, 124 | mpjaodan 772 | . . . 4 |
126 | 11, 40, 67, 125 | f1o2d 5943 | . . 3 |
127 | f1oeng 6619 | . . 3 | |
128 | 10, 126, 127 | syl2anc 408 | . 2 |
129 | fidceq 6731 | . . 3 DECID | |
130 | exmiddc 806 | . . 3 DECID | |
131 | 129, 130 | syl 14 | . 2 |
132 | 9, 128, 131 | mpjaodan 772 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 DECID wdc 804 w3a 947 wceq 1316 wcel 1465 wne 2285 cvv 2660 cdif 3038 cif 3444 csn 3497 class class class wbr 3899 cmpt 3959 wf1o 5092 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-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-en 6603 df-fin 6605 |
This theorem is referenced by: dif1en 6741 |
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