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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 6648 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | 3ad2ant1 1002 | . 2 |
4 | bren 6634 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 5 | ad2antll 482 | . . 3 |
7 | f1of 5360 | . . . . . . . . . 10 | |
8 | 7 | adantl 275 | . . . . . . . . 9 |
9 | simpll2 1021 | . . . . . . . . 9 | |
10 | 8, 9 | ffvelrnd 5549 | . . . . . . . 8 |
11 | simplrl 524 | . . . . . . . 8 | |
12 | elnn 4514 | . . . . . . . 8 | |
13 | 10, 11, 12 | syl2anc 408 | . . . . . . 7 |
14 | simpll3 1022 | . . . . . . . . 9 | |
15 | 8, 14 | ffvelrnd 5549 | . . . . . . . 8 |
16 | elnn 4514 | . . . . . . . 8 | |
17 | 15, 11, 16 | syl2anc 408 | . . . . . . 7 |
18 | nndceq 6388 | . . . . . . 7 DECID | |
19 | 13, 17, 18 | syl2anc 408 | . . . . . 6 DECID |
20 | exmiddc 821 | . . . . . 6 DECID | |
21 | 19, 20 | syl 14 | . . . . 5 |
22 | f1of1 5359 | . . . . . . . 8 | |
23 | 22 | adantl 275 | . . . . . . 7 |
24 | f1veqaeq 5663 | . . . . . . 7 | |
25 | 23, 9, 14, 24 | syl12anc 1214 | . . . . . 6 |
26 | fveq2 5414 | . . . . . . . 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 2552 | 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 2415 class class class wbr 3924 com 4499 wf 5114 wf1 5115 wf1o 5117 cfv 5118 cen 6625 cfn 6627 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-en 6628 df-fin 6630 |
This theorem is referenced by: fidifsnen 6757 fidifsnid 6758 unfiexmid 6799 undiffi 6806 fidcenumlemim 6833 |
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