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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 6623 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | adantr 274 | . 2 |
4 | elex2 2676 | . . . . . . . . 9 | |
5 | 4 | adantl 275 | . . . . . . . 8 |
6 | fin0 6747 | . . . . . . . . 9 | |
7 | 6 | adantr 274 | . . . . . . . 8 |
8 | 5, 7 | mpbird 166 | . . . . . . 7 |
9 | 8 | adantr 274 | . . . . . 6 |
10 | 9 | neneqd 2306 | . . . . 5 |
11 | simplrr 510 | . . . . . . 7 | |
12 | en0 6657 | . . . . . . . . 9 | |
13 | 12 | biimpri 132 | . . . . . . . 8 |
14 | 13 | adantl 275 | . . . . . . 7 |
15 | entr 6646 | . . . . . . 7 | |
16 | 11, 14, 15 | syl2anc 408 | . . . . . 6 |
17 | en0 6657 | . . . . . 6 | |
18 | 16, 17 | sylib 121 | . . . . 5 |
19 | 10, 18 | mtand 639 | . . . 4 |
20 | nn0suc 4488 | . . . . . 6 | |
21 | 20 | orcomd 703 | . . . . 5 |
22 | 21 | ad2antrl 481 | . . . 4 |
23 | 19, 22 | ecased 1312 | . . 3 |
24 | nnfi 6734 | . . . . 5 | |
25 | 24 | ad2antrl 481 | . . . 4 |
26 | simprl 505 | . . . . 5 | |
27 | simplrr 510 | . . . . . 6 | |
28 | breq2 3903 | . . . . . . 7 | |
29 | 28 | ad2antll 482 | . . . . . 6 |
30 | 27, 29 | mpbid 146 | . . . . 5 |
31 | simpllr 508 | . . . . 5 | |
32 | dif1en 6741 | . . . . 5 | |
33 | 26, 30, 31, 32 | syl3anc 1201 | . . . 4 |
34 | enfii 6736 | . . . 4 | |
35 | 25, 33, 34 | syl2anc 408 | . . 3 |
36 | 23, 35 | rexlimddv 2531 | . 2 |
37 | 3, 36 | rexlimddv 2531 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 wceq 1316 wex 1453 wcel 1465 wne 2285 wrex 2394 cdif 3038 c0 3333 csn 3497 class class class wbr 3899 csuc 4257 com 4474 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-fal 1322 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-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: diffifi 6756 zfz1isolemsplit 10549 zfz1isolem1 10551 fsumdifsnconst 11192 |
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