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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 4105 | . . . . . 6 | |
2 | 1 | 3ad2ant1 1003 | . . . . 5 |
3 | 2 | adantr 274 | . . . 4 |
4 | enrefg 6706 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | sneq 3571 | . . . . 5 | |
7 | 6 | difeq2d 3225 | . . . 4 |
8 | 7 | adantl 275 | . . 3 |
9 | 5, 8 | breqtrd 3990 | . 2 |
10 | 2 | adantr 274 | . . 3 |
11 | eqid 2157 | . . . 4 | |
12 | iftrue 3510 | . . . . . . . 8 | |
13 | 12 | adantl 275 | . . . . . . 7 |
14 | simpll2 1022 | . . . . . . . 8 | |
15 | 14 | adantr 274 | . . . . . . 7 |
16 | 13, 15 | eqeltrd 2234 | . . . . . 6 |
17 | simpllr 524 | . . . . . . . 8 | |
18 | 13 | eqeq1d 2166 | . . . . . . . 8 |
19 | 17, 18 | mtbird 663 | . . . . . . 7 |
20 | 19 | neneqad 2406 | . . . . . 6 |
21 | eldifsn 3686 | . . . . . 6 | |
22 | 16, 20, 21 | sylanbrc 414 | . . . . 5 |
23 | iffalse 3513 | . . . . . . . 8 | |
24 | 23 | adantl 275 | . . . . . . 7 |
25 | eldifi 3229 | . . . . . . . 8 | |
26 | 25 | ad2antlr 481 | . . . . . . 7 |
27 | 24, 26 | eqeltrd 2234 | . . . . . 6 |
28 | simpr 109 | . . . . . . . 8 | |
29 | 24 | eqeq1d 2166 | . . . . . . . 8 |
30 | 28, 29 | mtbird 663 | . . . . . . 7 |
31 | 30 | neneqad 2406 | . . . . . 6 |
32 | 27, 31, 21 | sylanbrc 414 | . . . . 5 |
33 | simpll1 1021 | . . . . . . 7 | |
34 | 25 | adantl 275 | . . . . . . 7 |
35 | simpll3 1023 | . . . . . . 7 | |
36 | fidceq 6811 | . . . . . . 7 DECID | |
37 | 33, 34, 35, 36 | syl3anc 1220 | . . . . . 6 DECID |
38 | exmiddc 822 | . . . . . 6 DECID | |
39 | 37, 38 | syl 14 | . . . . 5 |
40 | 22, 32, 39 | mpjaodan 788 | . . . 4 |
41 | iftrue 3510 | . . . . . . 7 | |
42 | 41 | adantl 275 | . . . . . 6 |
43 | simpl3 987 | . . . . . . . 8 | |
44 | simpr 109 | . . . . . . . . . 10 | |
45 | 44 | neneqad 2406 | . . . . . . . . 9 |
46 | 45 | necomd 2413 | . . . . . . . 8 |
47 | eldifsn 3686 | . . . . . . . 8 | |
48 | 43, 46, 47 | sylanbrc 414 | . . . . . . 7 |
49 | 48 | ad2antrr 480 | . . . . . 6 |
50 | 42, 49 | eqeltrd 2234 | . . . . 5 |
51 | iffalse 3513 | . . . . . . 7 | |
52 | 51 | adantl 275 | . . . . . 6 |
53 | eldifi 3229 | . . . . . . . 8 | |
54 | 53 | ad2antlr 481 | . . . . . . 7 |
55 | simpr 109 | . . . . . . . 8 | |
56 | 55 | neneqad 2406 | . . . . . . 7 |
57 | eldifsn 3686 | . . . . . . 7 | |
58 | 54, 56, 57 | sylanbrc 414 | . . . . . 6 |
59 | 52, 58 | eqeltrd 2234 | . . . . 5 |
60 | simpll1 1021 | . . . . . . 7 | |
61 | 53 | adantl 275 | . . . . . . 7 |
62 | simpll2 1022 | . . . . . . 7 | |
63 | fidceq 6811 | . . . . . . 7 DECID | |
64 | 60, 61, 62, 63 | syl3anc 1220 | . . . . . 6 DECID |
65 | exmiddc 822 | . . . . . 6 DECID | |
66 | 64, 65 | syl 14 | . . . . 5 |
67 | 50, 59, 66 | mpjaodan 788 | . . . 4 |
68 | 12 | adantl 275 | . . . . . . . . . 10 |
69 | 68 | eqeq2d 2169 | . . . . . . . . 9 |
70 | 69 | biimpar 295 | . . . . . . . 8 |
71 | 70 | a1d 22 | . . . . . . 7 |
72 | simpr 109 | . . . . . . . . . . 11 | |
73 | 51 | eqeq2d 2169 | . . . . . . . . . . . 12 |
74 | 73 | ad2antlr 481 | . . . . . . . . . . 11 |
75 | 72, 74 | mpbid 146 | . . . . . . . . . 10 |
76 | simpllr 524 | . . . . . . . . . 10 | |
77 | 75, 76 | eqtr3d 2192 | . . . . . . . . 9 |
78 | simprr 522 | . . . . . . . . . . . . 13 | |
79 | 78 | ad2antrr 480 | . . . . . . . . . . . 12 |
80 | 79 | eldifbd 3114 | . . . . . . . . . . 11 |
81 | 80 | adantr 274 | . . . . . . . . . 10 |
82 | velsn 3577 | . . . . . . . . . 10 | |
83 | 81, 82 | sylnib 666 | . . . . . . . . 9 |
84 | 77, 83 | pm2.21dd 610 | . . . . . . . 8 |
85 | 84 | ex 114 | . . . . . . 7 |
86 | simpll1 1021 | . . . . . . . . . 10 | |
87 | 53 | ad2antll 483 | . . . . . . . . . 10 |
88 | simpll2 1022 | . . . . . . . . . 10 | |
89 | 86, 87, 88, 63 | syl3anc 1220 | . . . . . . . . 9 DECID |
90 | 89, 65 | syl 14 | . . . . . . . 8 |
91 | 90 | adantr 274 | . . . . . . 7 |
92 | 71, 85, 91 | mpjaodan 788 | . . . . . 6 |
93 | 41 | eqeq2d 2169 | . . . . . . . . 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 520 | . . . . . . . . 9 | |
99 | 41 | adantl 275 | . . . . . . . . 9 |
100 | 98, 99 | eqtrd 2190 | . . . . . . . 8 |
101 | simpllr 524 | . . . . . . . 8 | |
102 | 100, 101 | pm2.21dd 610 | . . . . . . 7 |
103 | 23 | ad3antlr 485 | . . . . . . . 8 |
104 | simplr 520 | . . . . . . . . 9 | |
105 | 51 | adantl 275 | . . . . . . . . 9 |
106 | 104, 105 | eqtrd 2190 | . . . . . . . 8 |
107 | 103, 106 | eqtr2d 2191 | . . . . . . 7 |
108 | 90 | ad2antrr 480 | . . . . . . 7 |
109 | 102, 107, 108 | mpjaodan 788 | . . . . . 6 |
110 | simprl 521 | . . . . . . . . . . . 12 | |
111 | 110 | eldifbd 3114 | . . . . . . . . . . 11 |
112 | velsn 3577 | . . . . . . . . . . 11 | |
113 | 111, 112 | sylnib 666 | . . . . . . . . . 10 |
114 | 113 | ad2antrr 480 | . . . . . . . . 9 |
115 | simpr 109 | . . . . . . . . . . 11 | |
116 | 23 | eqeq2d 2169 | . . . . . . . . . . . 12 |
117 | 116 | ad2antlr 481 | . . . . . . . . . . 11 |
118 | 115, 117 | mpbid 146 | . . . . . . . . . 10 |
119 | 118 | eqeq1d 2166 | . . . . . . . . 9 |
120 | 114, 119 | mtbird 663 | . . . . . . . 8 |
121 | 120, 51 | syl 14 | . . . . . . 7 |
122 | 121, 118 | eqtr2d 2191 | . . . . . 6 |
123 | 109, 122 | impbida 586 | . . . . 5 |
124 | 39 | adantrr 471 | . . . . 5 |
125 | 97, 123, 124 | mpjaodan 788 | . . . 4 |
126 | 11, 40, 67, 125 | f1o2d 6022 | . . 3 |
127 | f1oeng 6699 | . . 3 | |
128 | 10, 126, 127 | syl2anc 409 | . 2 |
129 | fidceq 6811 | . . 3 DECID | |
130 | exmiddc 822 | . . 3 DECID | |
131 | 129, 130 | syl 14 | . 2 |
132 | 9, 128, 131 | mpjaodan 788 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 820 w3a 963 wceq 1335 wcel 2128 wne 2327 cvv 2712 cdif 3099 cif 3505 csn 3560 class class class wbr 3965 cmpt 4025 wf1o 5168 cen 6680 cfn 6682 |
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 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 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 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-en 6683 df-fin 6685 |
This theorem is referenced by: dif1en 6821 |
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