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Mirrors > Home > ILE Home > Th. List > difinfinf | Unicode version |
Description: An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.) |
Ref | Expression |
---|---|
difinfinf | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq2 3229 | . . 3 | |
2 | 1 | breq2d 3988 | . 2 |
3 | difeq2 3229 | . . 3 | |
4 | 3 | breq2d 3988 | . 2 |
5 | difeq2 3229 | . . 3 | |
6 | 5 | breq2d 3988 | . 2 |
7 | difeq2 3229 | . . 3 | |
8 | 7 | breq2d 3988 | . 2 |
9 | simplr 520 | . . 3 DECID | |
10 | dif0 3474 | . . 3 | |
11 | 9, 10 | breqtrrdi 4018 | . 2 DECID |
12 | difss 3243 | . . . . . . 7 | |
13 | ssralv 3201 | . . . . . . . . 9 DECID DECID | |
14 | 12, 13 | ax-mp 5 | . . . . . . . 8 DECID DECID |
15 | 14 | ralimi 2527 | . . . . . . 7 DECID DECID |
16 | ssralv 3201 | . . . . . . 7 DECID DECID | |
17 | 12, 15, 16 | mpsyl 65 | . . . . . 6 DECID DECID |
18 | 17 | ad5antr 488 | . . . . 5 DECID DECID |
19 | simpr 109 | . . . . 5 DECID | |
20 | simprl 521 | . . . . . . 7 DECID | |
21 | 20 | ad3antrrr 484 | . . . . . 6 DECID |
22 | simplrr 526 | . . . . . 6 DECID | |
23 | ssdif 3252 | . . . . . . 7 | |
24 | 23 | sseld 3136 | . . . . . 6 |
25 | 21, 22, 24 | sylc 62 | . . . . 5 DECID |
26 | difinfsn 7056 | . . . . 5 DECID | |
27 | 18, 19, 25, 26 | syl3anc 1227 | . . . 4 DECID |
28 | difun1 3377 | . . . 4 | |
29 | 27, 28 | breqtrrdi 4018 | . . 3 DECID |
30 | 29 | ex 114 | . 2 DECID |
31 | simprr 522 | . 2 DECID | |
32 | 2, 4, 6, 8, 11, 30, 31 | findcard2sd 6849 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 824 wceq 1342 wcel 2135 wral 2442 cdif 3108 cun 3109 wss 3111 c0 3404 csn 3570 class class class wbr 3976 com 4561 cdom 6696 cfn 6697 |
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-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 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-if 3516 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-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-dju 6994 df-inl 7003 df-inr 7004 df-case 7040 |
This theorem is referenced by: inffinp1 12305 |
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