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Mirrors > Home > ILE Home > Th. List > ctssdclemr | Unicode version |
Description: Lemma for ctssdc 7069. 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 5400 | . . 3 ⊔ ⊔ | |
2 | 1 | cbvexv 1905 | . 2 ⊔ ⊔ |
3 | id 19 | . . . . . 6 ⊔ ⊔ | |
4 | eqid 2164 | . . . . . 6 inl inl | |
5 | eqid 2164 | . . . . . 6 inl inl | |
6 | 3, 4, 5 | ctssdccl 7067 | . . . . 5 ⊔ inl inl inl DECID inl |
7 | djulf1o 7014 | . . . . . . . . 9 inl | |
8 | f1ocnv 5439 | . . . . . . . . 9 inl inl | |
9 | f1ofun 5428 | . . . . . . . . 9 inl inl | |
10 | 7, 8, 9 | mp2b 8 | . . . . . . . 8 inl |
11 | vex 2724 | . . . . . . . 8 | |
12 | cofunexg 6071 | . . . . . . . 8 inl inl | |
13 | 10, 11, 12 | mp2an 423 | . . . . . . 7 inl |
14 | foeq1 5400 | . . . . . . 7 inl inl inl inl | |
15 | 13, 14 | spcev 2816 | . . . . . 6 inl inl inl |
16 | 15 | 3anim2i 1175 | . . . . 5 inl inl inl DECID inl inl inl DECID inl |
17 | 6, 16 | syl 14 | . . . 4 ⊔ inl inl DECID inl |
18 | omex 4564 | . . . . . 6 | |
19 | 18 | rabex 4120 | . . . . 5 inl |
20 | sseq1 3160 | . . . . . 6 inl inl | |
21 | foeq2 5401 | . . . . . . 7 inl inl | |
22 | 21 | exbidv 1812 | . . . . . 6 inl inl |
23 | eleq2 2228 | . . . . . . . 8 inl inl | |
24 | 23 | dcbid 828 | . . . . . . 7 inl DECID DECID inl |
25 | 24 | ralbidv 2464 | . . . . . 6 inl DECID DECID inl |
26 | 20, 22, 25 | 3anbi123d 1301 | . . . . 5 inl DECID inl inl DECID inl |
27 | 19, 26 | spcev 2816 | . . . 4 inl inl DECID inl DECID |
28 | 17, 27 | syl 14 | . . 3 ⊔ DECID |
29 | 28 | exlimiv 1585 | . 2 ⊔ DECID |
30 | 2, 29 | sylbi 120 | 1 ⊔ DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 DECID wdc 824 w3a 967 wceq 1342 wex 1479 wcel 2135 wral 2442 crab 2446 cvv 2721 wss 3111 c0 3404 csn 3570 com 4561 cxp 4596 ccnv 4597 cima 4601 ccom 4602 wfun 5176 wfo 5180 wf1o 5181 cfv 5182 c1o 6368 ⊔ cdju 6993 inlcinl 7001 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1st 6100 df-2nd 6101 df-1o 6375 df-dju 6994 df-inl 7003 df-inr 7004 |
This theorem is referenced by: ctssdc 7069 |
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