Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > enumct | Unicode version |
Description: A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen], p. 73 (except that our definition of countable does not require the set to be inhabited). "Finitely enumerable" is defined as per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as ⊔ per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.) |
Ref | Expression |
---|---|
enumct | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 518 | . . . . . . . . 9 | |
2 | foeq2 5342 | . . . . . . . . . 10 | |
3 | 2 | adantl 275 | . . . . . . . . 9 |
4 | 1, 3 | mpbid 146 | . . . . . . . 8 |
5 | fo00 5403 | . . . . . . . 8 | |
6 | 4, 5 | sylib 121 | . . . . . . 7 |
7 | 0ct 6992 | . . . . . . . 8 ⊔ | |
8 | djueq1 6925 | . . . . . . . . . 10 ⊔ ⊔ | |
9 | foeq3 5343 | . . . . . . . . . 10 ⊔ ⊔ ⊔ ⊔ | |
10 | 8, 9 | syl 14 | . . . . . . . . 9 ⊔ ⊔ |
11 | 10 | exbidv 1797 | . . . . . . . 8 ⊔ ⊔ |
12 | 7, 11 | mpbiri 167 | . . . . . . 7 ⊔ |
13 | 6, 12 | simpl2im 383 | . . . . . 6 ⊔ |
14 | omex 4507 | . . . . . . . . 9 | |
15 | 14 | mptex 5646 | . . . . . . . 8 |
16 | simpll 518 | . . . . . . . . 9 | |
17 | simplr 519 | . . . . . . . . 9 | |
18 | simpr 109 | . . . . . . . . 9 | |
19 | eqid 2139 | . . . . . . . . 9 | |
20 | 16, 17, 18, 19 | enumctlemm 6999 | . . . . . . . 8 |
21 | foeq1 5341 | . . . . . . . . 9 | |
22 | 21 | spcegv 2774 | . . . . . . . 8 |
23 | 15, 20, 22 | mpsyl 65 | . . . . . . 7 |
24 | fof 5345 | . . . . . . . . . . 11 | |
25 | 24 | ad2antrr 479 | . . . . . . . . . 10 |
26 | 25, 18 | ffvelrnd 5556 | . . . . . . . . 9 |
27 | eleq1 2202 | . . . . . . . . . 10 | |
28 | 27 | spcegv 2774 | . . . . . . . . 9 |
29 | 26, 26, 28 | sylc 62 | . . . . . . . 8 |
30 | ctm 6994 | . . . . . . . 8 ⊔ | |
31 | 29, 30 | syl 14 | . . . . . . 7 ⊔ |
32 | 23, 31 | mpbird 166 | . . . . . 6 ⊔ |
33 | 0elnn 4532 | . . . . . . 7 | |
34 | 33 | adantl 275 | . . . . . 6 |
35 | 13, 32, 34 | mpjaodan 787 | . . . . 5 ⊔ |
36 | 35 | ex 114 | . . . 4 ⊔ |
37 | 36 | exlimiv 1577 | . . 3 ⊔ |
38 | 37 | impcom 124 | . 2 ⊔ |
39 | 38 | rexlimiva 2544 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wex 1468 wcel 1480 wrex 2417 cvv 2686 c0 3363 cif 3474 cmpt 3989 com 4504 wf 5119 wfo 5121 cfv 5123 c1o 6306 ⊔ cdju 6922 |
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-1st 6038 df-2nd 6039 df-1o 6313 df-dju 6923 df-inl 6932 df-inr 6933 df-case 6969 |
This theorem is referenced by: finct 7001 |
Copyright terms: Public domain | W3C validator |