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Mirrors > Home > ILE Home > Th. List > diffisn | Unicode version |
Description: Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.) |
Ref | Expression |
---|---|
diffisn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6264 | . . . 4 | |
2 | 1 | biimpi 118 | . . 3 |
3 | 2 | adantr 270 | . 2 |
4 | elex2 2615 | . . . . . . . . 9 | |
5 | 4 | adantl 271 | . . . . . . . 8 |
6 | fin0 6369 | . . . . . . . . 9 | |
7 | 6 | adantr 270 | . . . . . . . 8 |
8 | 5, 7 | mpbird 165 | . . . . . . 7 |
9 | 8 | adantr 270 | . . . . . 6 |
10 | 9 | neneqd 2266 | . . . . 5 |
11 | simplrr 502 | . . . . . . 7 | |
12 | en0 6298 | . . . . . . . . 9 | |
13 | 12 | biimpri 131 | . . . . . . . 8 |
14 | 13 | adantl 271 | . . . . . . 7 |
15 | entr 6287 | . . . . . . 7 | |
16 | 11, 14, 15 | syl2anc 403 | . . . . . 6 |
17 | en0 6298 | . . . . . 6 | |
18 | 16, 17 | sylib 120 | . . . . 5 |
19 | 10, 18 | mtand 623 | . . . 4 |
20 | nn0suc 4345 | . . . . . 6 | |
21 | 20 | orcomd 680 | . . . . 5 |
22 | 21 | ad2antrl 473 | . . . 4 |
23 | 19, 22 | ecased 1280 | . . 3 |
24 | nnfi 6357 | . . . . 5 | |
25 | 24 | ad2antrl 473 | . . . 4 |
26 | simprl 497 | . . . . 5 | |
27 | simplrr 502 | . . . . . 6 | |
28 | breq2 3789 | . . . . . . 7 | |
29 | 28 | ad2antll 474 | . . . . . 6 |
30 | 27, 29 | mpbid 145 | . . . . 5 |
31 | simpllr 500 | . . . . 5 | |
32 | dif1en 6364 | . . . . 5 | |
33 | 26, 30, 31, 32 | syl3anc 1169 | . . . 4 |
34 | enfii 6359 | . . . 4 | |
35 | 25, 33, 34 | syl2anc 403 | . . 3 |
36 | 23, 35 | rexlimddv 2481 | . 2 |
37 | 3, 36 | rexlimddv 2481 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 wceq 1284 wex 1421 wcel 1433 wne 2245 wrex 2349 cdif 2970 c0 3251 csn 3398 class class class wbr 3785 csuc 4120 com 4331 cen 6242 cfn 6244 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-er 6129 df-en 6245 df-fin 6247 |
This theorem is referenced by: diffifi 6378 |
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