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Mirrors > Home > ILE Home > Th. List > diffifi | Unicode version |
Description: Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.) |
Ref | Expression |
---|---|
diffifi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 987 | . 2 | |
2 | simp1 986 | . 2 | |
3 | simp3 988 | . 2 | |
4 | sseq1 3161 | . . . . . 6 | |
5 | 4 | anbi2d 460 | . . . . 5 |
6 | difeq2 3230 | . . . . . 6 | |
7 | 6 | eleq1d 2233 | . . . . 5 |
8 | 5, 7 | imbi12d 233 | . . . 4 |
9 | sseq1 3161 | . . . . . 6 | |
10 | 9 | anbi2d 460 | . . . . 5 |
11 | difeq2 3230 | . . . . . 6 | |
12 | 11 | eleq1d 2233 | . . . . 5 |
13 | 10, 12 | imbi12d 233 | . . . 4 |
14 | sseq1 3161 | . . . . . 6 | |
15 | 14 | anbi2d 460 | . . . . 5 |
16 | difeq2 3230 | . . . . . 6 | |
17 | 16 | eleq1d 2233 | . . . . 5 |
18 | 15, 17 | imbi12d 233 | . . . 4 |
19 | sseq1 3161 | . . . . . 6 | |
20 | 19 | anbi2d 460 | . . . . 5 |
21 | difeq2 3230 | . . . . . 6 | |
22 | 21 | eleq1d 2233 | . . . . 5 |
23 | 20, 22 | imbi12d 233 | . . . 4 |
24 | dif0 3475 | . . . . . . 7 | |
25 | 24 | eleq1i 2230 | . . . . . 6 |
26 | 25 | biimpri 132 | . . . . 5 |
27 | 26 | adantr 274 | . . . 4 |
28 | difun1 3378 | . . . . . 6 | |
29 | simprl 521 | . . . . . . . 8 | |
30 | simprr 522 | . . . . . . . . 9 | |
31 | 30 | unssad 3295 | . . . . . . . 8 |
32 | simplr 520 | . . . . . . . 8 | |
33 | 29, 31, 32 | mp2and 430 | . . . . . . 7 |
34 | vsnid 3603 | . . . . . . . . . 10 | |
35 | simprr 522 | . . . . . . . . . . . 12 | |
36 | 35 | unssbd 3296 | . . . . . . . . . . 11 |
37 | 36 | sseld 3137 | . . . . . . . . . 10 |
38 | 34, 37 | mpi 15 | . . . . . . . . 9 |
39 | 38 | adantllr 473 | . . . . . . . 8 |
40 | simpllr 524 | . . . . . . . 8 | |
41 | 39, 40 | eldifd 3122 | . . . . . . 7 |
42 | diffisn 6851 | . . . . . . 7 | |
43 | 33, 41, 42 | syl2anc 409 | . . . . . 6 |
44 | 28, 43 | eqeltrid 2251 | . . . . 5 |
45 | 44 | exp31 362 | . . . 4 |
46 | 8, 13, 18, 23, 27, 45 | findcard2s 6848 | . . 3 |
47 | 46 | imp 123 | . 2 |
48 | 1, 2, 3, 47 | syl12anc 1225 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 cdif 3109 cun 3110 wss 3112 c0 3405 csn 3571 cfn 6698 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-if 3517 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-iord 4339 df-on 4341 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-er 6493 df-en 6699 df-fin 6701 |
This theorem is referenced by: unfiin 6883 fihashssdif 10721 hashdifpr 10723 fsumlessfi 11391 hash2iun1dif1 11411 |
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