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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 905 | . 2 | |
2 | simp1 904 | . 2 | |
3 | simp3 906 | . 2 | |
4 | sseq1 2966 | . . . . . 6 | |
5 | 4 | anbi2d 437 | . . . . 5 |
6 | difeq2 3056 | . . . . . 6 | |
7 | 6 | eleq1d 2106 | . . . . 5 |
8 | 5, 7 | imbi12d 223 | . . . 4 |
9 | sseq1 2966 | . . . . . 6 | |
10 | 9 | anbi2d 437 | . . . . 5 |
11 | difeq2 3056 | . . . . . 6 | |
12 | 11 | eleq1d 2106 | . . . . 5 |
13 | 10, 12 | imbi12d 223 | . . . 4 |
14 | sseq1 2966 | . . . . . 6 | |
15 | 14 | anbi2d 437 | . . . . 5 |
16 | difeq2 3056 | . . . . . 6 | |
17 | 16 | eleq1d 2106 | . . . . 5 |
18 | 15, 17 | imbi12d 223 | . . . 4 |
19 | sseq1 2966 | . . . . . 6 | |
20 | 19 | anbi2d 437 | . . . . 5 |
21 | difeq2 3056 | . . . . . 6 | |
22 | 21 | eleq1d 2106 | . . . . 5 |
23 | 20, 22 | imbi12d 223 | . . . 4 |
24 | dif0 3294 | . . . . . . 7 | |
25 | 24 | eleq1i 2103 | . . . . . 6 |
26 | 25 | biimpri 124 | . . . . 5 |
27 | 26 | adantr 261 | . . . 4 |
28 | difun1 3197 | . . . . . 6 | |
29 | simprl 483 | . . . . . . . 8 | |
30 | simprr 484 | . . . . . . . . 9 | |
31 | 30 | unssad 3120 | . . . . . . . 8 |
32 | simplr 482 | . . . . . . . 8 | |
33 | 29, 31, 32 | mp2and 409 | . . . . . . 7 |
34 | vsnid 3403 | . . . . . . . . . 10 | |
35 | simprr 484 | . . . . . . . . . . . 12 | |
36 | 35 | unssbd 3121 | . . . . . . . . . . 11 |
37 | 36 | sseld 2944 | . . . . . . . . . 10 |
38 | 34, 37 | mpi 15 | . . . . . . . . 9 |
39 | 38 | adantllr 450 | . . . . . . . 8 |
40 | simpllr 486 | . . . . . . . 8 | |
41 | 39, 40 | eldifd 2928 | . . . . . . 7 |
42 | diffisn 6350 | . . . . . . 7 | |
43 | 33, 41, 42 | syl2anc 391 | . . . . . 6 |
44 | 28, 43 | syl5eqel 2124 | . . . . 5 |
45 | 44 | exp31 346 | . . . 4 |
46 | 8, 13, 18, 23, 27, 45 | findcard2s 6347 | . . 3 |
47 | 46 | imp 115 | . 2 |
48 | 1, 2, 3, 47 | syl12anc 1133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 cdif 2914 cun 2915 wss 2917 c0 3224 csn 3375 cfn 6221 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-er 6106 df-en 6222 df-fin 6224 |
This theorem is referenced by: (None) |
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