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Mirrors > Home > ILE Home > Th. List > fidceq | Unicode version |
Description: Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that is finite would require showing it is equinumerous to or to but to show that you'd need to know or , respectively. (Contributed by Jim Kingdon, 5-Sep-2021.) |
Ref | Expression |
---|---|
fidceq | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6655 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | 3ad2ant1 1002 | . 2 |
4 | bren 6641 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 5 | ad2antll 482 | . . 3 |
7 | f1of 5367 | . . . . . . . . . 10 | |
8 | 7 | adantl 275 | . . . . . . . . 9 |
9 | simpll2 1021 | . . . . . . . . 9 | |
10 | 8, 9 | ffvelrnd 5556 | . . . . . . . 8 |
11 | simplrl 524 | . . . . . . . 8 | |
12 | elnn 4519 | . . . . . . . 8 | |
13 | 10, 11, 12 | syl2anc 408 | . . . . . . 7 |
14 | simpll3 1022 | . . . . . . . . 9 | |
15 | 8, 14 | ffvelrnd 5556 | . . . . . . . 8 |
16 | elnn 4519 | . . . . . . . 8 | |
17 | 15, 11, 16 | syl2anc 408 | . . . . . . 7 |
18 | nndceq 6395 | . . . . . . 7 DECID | |
19 | 13, 17, 18 | syl2anc 408 | . . . . . 6 DECID |
20 | exmiddc 821 | . . . . . 6 DECID | |
21 | 19, 20 | syl 14 | . . . . 5 |
22 | f1of1 5366 | . . . . . . . 8 | |
23 | 22 | adantl 275 | . . . . . . 7 |
24 | f1veqaeq 5670 | . . . . . . 7 | |
25 | 23, 9, 14, 24 | syl12anc 1214 | . . . . . 6 |
26 | fveq2 5421 | . . . . . . . 8 | |
27 | 26 | con3i 621 | . . . . . . 7 |
28 | 27 | a1i 9 | . . . . . 6 |
29 | 25, 28 | orim12d 775 | . . . . 5 |
30 | 21, 29 | mpd 13 | . . . 4 |
31 | df-dc 820 | . . . 4 DECID | |
32 | 30, 31 | sylibr 133 | . . 3 DECID |
33 | 6, 32 | exlimddv 1870 | . 2 DECID |
34 | 3, 33 | rexlimddv 2554 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 w3a 962 wceq 1331 wex 1468 wcel 1480 wrex 2417 class class class wbr 3929 com 4504 wf 5119 wf1 5120 wf1o 5122 cfv 5123 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-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-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-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 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-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-en 6635 df-fin 6637 |
This theorem is referenced by: fidifsnen 6764 fidifsnid 6765 unfiexmid 6806 undiffi 6813 fidcenumlemim 6840 |
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