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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 3239 | . . 3 | |
2 | 1 | breq2d 4001 | . 2 |
3 | difeq2 3239 | . . 3 | |
4 | 3 | breq2d 4001 | . 2 |
5 | difeq2 3239 | . . 3 | |
6 | 5 | breq2d 4001 | . 2 |
7 | difeq2 3239 | . . 3 | |
8 | 7 | breq2d 4001 | . 2 |
9 | simplr 525 | . . 3 DECID | |
10 | dif0 3485 | . . 3 | |
11 | 9, 10 | breqtrrdi 4031 | . 2 DECID |
12 | difss 3253 | . . . . . . 7 | |
13 | ssralv 3211 | . . . . . . . . 9 DECID DECID | |
14 | 12, 13 | ax-mp 5 | . . . . . . . 8 DECID DECID |
15 | 14 | ralimi 2533 | . . . . . . 7 DECID DECID |
16 | ssralv 3211 | . . . . . . 7 DECID DECID | |
17 | 12, 15, 16 | mpsyl 65 | . . . . . 6 DECID DECID |
18 | 17 | ad5antr 493 | . . . . 5 DECID DECID |
19 | simpr 109 | . . . . 5 DECID | |
20 | simprl 526 | . . . . . . 7 DECID | |
21 | 20 | ad3antrrr 489 | . . . . . 6 DECID |
22 | simplrr 531 | . . . . . 6 DECID | |
23 | ssdif 3262 | . . . . . . 7 | |
24 | 23 | sseld 3146 | . . . . . 6 |
25 | 21, 22, 24 | sylc 62 | . . . . 5 DECID |
26 | difinfsn 7077 | . . . . 5 DECID | |
27 | 18, 19, 25, 26 | syl3anc 1233 | . . . 4 DECID |
28 | difun1 3387 | . . . 4 | |
29 | 27, 28 | breqtrrdi 4031 | . . 3 DECID |
30 | 29 | ex 114 | . 2 DECID |
31 | simprr 527 | . 2 DECID | |
32 | 2, 4, 6, 8, 11, 30, 31 | findcard2sd 6870 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 829 wceq 1348 wcel 2141 wral 2448 cdif 3118 cun 3119 wss 3121 c0 3414 csn 3583 class class class wbr 3989 com 4574 cdom 6717 cfn 6718 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-1st 6119 df-2nd 6120 df-1o 6395 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-dju 7015 df-inl 7024 df-inr 7025 df-case 7061 |
This theorem is referenced by: inffinp1 12384 |
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