Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ctssdclemr | Unicode version |
Description: Lemma for ctssdc 6998. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.) |
Ref | Expression |
---|---|
ctssdclemr | ⊔ DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | foeq1 5341 | . . 3 ⊔ ⊔ | |
2 | 1 | cbvexv 1890 | . 2 ⊔ ⊔ |
3 | id 19 | . . . . . 6 ⊔ ⊔ | |
4 | eqid 2139 | . . . . . 6 inl inl | |
5 | eqid 2139 | . . . . . 6 inl inl | |
6 | 3, 4, 5 | ctssdccl 6996 | . . . . 5 ⊔ inl inl inl DECID inl |
7 | djulf1o 6943 | . . . . . . . . 9 inl | |
8 | f1ocnv 5380 | . . . . . . . . 9 inl inl | |
9 | f1ofun 5369 | . . . . . . . . 9 inl inl | |
10 | 7, 8, 9 | mp2b 8 | . . . . . . . 8 inl |
11 | vex 2689 | . . . . . . . 8 | |
12 | cofunexg 6009 | . . . . . . . 8 inl inl | |
13 | 10, 11, 12 | mp2an 422 | . . . . . . 7 inl |
14 | foeq1 5341 | . . . . . . 7 inl inl inl inl | |
15 | 13, 14 | spcev 2780 | . . . . . 6 inl inl inl |
16 | 15 | 3anim2i 1168 | . . . . 5 inl inl inl DECID inl inl inl DECID inl |
17 | 6, 16 | syl 14 | . . . 4 ⊔ inl inl DECID inl |
18 | omex 4507 | . . . . . 6 | |
19 | 18 | rabex 4072 | . . . . 5 inl |
20 | sseq1 3120 | . . . . . 6 inl inl | |
21 | foeq2 5342 | . . . . . . 7 inl inl | |
22 | 21 | exbidv 1797 | . . . . . 6 inl inl |
23 | eleq2 2203 | . . . . . . . 8 inl inl | |
24 | 23 | dcbid 823 | . . . . . . 7 inl DECID DECID inl |
25 | 24 | ralbidv 2437 | . . . . . 6 inl DECID DECID inl |
26 | 20, 22, 25 | 3anbi123d 1290 | . . . . 5 inl DECID inl inl DECID inl |
27 | 19, 26 | spcev 2780 | . . . 4 inl inl DECID inl DECID |
28 | 17, 27 | syl 14 | . . 3 ⊔ DECID |
29 | 28 | exlimiv 1577 | . 2 ⊔ DECID |
30 | 2, 29 | sylbi 120 | 1 ⊔ DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 DECID wdc 819 w3a 962 wceq 1331 wex 1468 wcel 1480 wral 2416 crab 2420 cvv 2686 wss 3071 c0 3363 csn 3527 com 4504 cxp 4537 ccnv 4538 cima 4542 ccom 4543 wfun 5117 wfo 5121 wf1o 5122 cfv 5123 c1o 6306 ⊔ cdju 6922 inlcinl 6930 |
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-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 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-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 |
This theorem is referenced by: ctssdc 6998 |
Copyright terms: Public domain | W3C validator |