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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 6655 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | adantr 274 | . 2 |
4 | elex2 2702 | . . . . . . . . 9 | |
5 | 4 | adantl 275 | . . . . . . . 8 |
6 | fin0 6779 | . . . . . . . . 9 | |
7 | 6 | adantr 274 | . . . . . . . 8 |
8 | 5, 7 | mpbird 166 | . . . . . . 7 |
9 | 8 | adantr 274 | . . . . . 6 |
10 | 9 | neneqd 2329 | . . . . 5 |
11 | simplrr 525 | . . . . . . 7 | |
12 | en0 6689 | . . . . . . . . 9 | |
13 | 12 | biimpri 132 | . . . . . . . 8 |
14 | 13 | adantl 275 | . . . . . . 7 |
15 | entr 6678 | . . . . . . 7 | |
16 | 11, 14, 15 | syl2anc 408 | . . . . . 6 |
17 | en0 6689 | . . . . . 6 | |
18 | 16, 17 | sylib 121 | . . . . 5 |
19 | 10, 18 | mtand 654 | . . . 4 |
20 | nn0suc 4518 | . . . . . 6 | |
21 | 20 | orcomd 718 | . . . . 5 |
22 | 21 | ad2antrl 481 | . . . 4 |
23 | 19, 22 | ecased 1327 | . . 3 |
24 | nnfi 6766 | . . . . 5 | |
25 | 24 | ad2antrl 481 | . . . 4 |
26 | simprl 520 | . . . . 5 | |
27 | simplrr 525 | . . . . . 6 | |
28 | breq2 3933 | . . . . . . 7 | |
29 | 28 | ad2antll 482 | . . . . . 6 |
30 | 27, 29 | mpbid 146 | . . . . 5 |
31 | simpllr 523 | . . . . 5 | |
32 | dif1en 6773 | . . . . 5 | |
33 | 26, 30, 31, 32 | syl3anc 1216 | . . . 4 |
34 | enfii 6768 | . . . 4 | |
35 | 25, 33, 34 | syl2anc 408 | . . 3 |
36 | 23, 35 | rexlimddv 2554 | . 2 |
37 | 3, 36 | rexlimddv 2554 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wex 1468 wcel 1480 wne 2308 wrex 2417 cdif 3068 c0 3363 csn 3527 class class class wbr 3929 csuc 4287 com 4504 cen 6632 cfn 6634 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: diffifi 6788 zfz1isolemsplit 10581 zfz1isolem1 10583 fsumdifsnconst 11224 |
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