Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3245 | . . 3 | |
2 | 1 | breq2d 4010 | . 2 |
3 | difeq2 3245 | . . 3 | |
4 | 3 | breq2d 4010 | . 2 |
5 | difeq2 3245 | . . 3 | |
6 | 5 | breq2d 4010 | . 2 |
7 | difeq2 3245 | . . 3 | |
8 | 7 | breq2d 4010 | . 2 |
9 | simplr 528 | . . 3 DECID | |
10 | dif0 3491 | . . 3 | |
11 | 9, 10 | breqtrrdi 4040 | . 2 DECID |
12 | difss 3259 | . . . . . . 7 | |
13 | ssralv 3217 | . . . . . . . . 9 DECID DECID | |
14 | 12, 13 | ax-mp 5 | . . . . . . . 8 DECID DECID |
15 | 14 | ralimi 2538 | . . . . . . 7 DECID DECID |
16 | ssralv 3217 | . . . . . . 7 DECID DECID | |
17 | 12, 15, 16 | mpsyl 65 | . . . . . 6 DECID DECID |
18 | 17 | ad5antr 496 | . . . . 5 DECID DECID |
19 | simpr 110 | . . . . 5 DECID | |
20 | simprl 529 | . . . . . . 7 DECID | |
21 | 20 | ad3antrrr 492 | . . . . . 6 DECID |
22 | simplrr 536 | . . . . . 6 DECID | |
23 | ssdif 3268 | . . . . . . 7 | |
24 | 23 | sseld 3152 | . . . . . 6 |
25 | 21, 22, 24 | sylc 62 | . . . . 5 DECID |
26 | difinfsn 7089 | . . . . 5 DECID | |
27 | 18, 19, 25, 26 | syl3anc 1238 | . . . 4 DECID |
28 | difun1 3393 | . . . 4 | |
29 | 27, 28 | breqtrrdi 4040 | . . 3 DECID |
30 | 29 | ex 115 | . 2 DECID |
31 | simprr 531 | . 2 DECID | |
32 | 2, 4, 6, 8, 11, 30, 31 | findcard2sd 6882 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 DECID wdc 834 wceq 1353 wcel 2146 wral 2453 cdif 3124 cun 3125 wss 3127 c0 3420 csn 3589 class class class wbr 3998 com 4583 cdom 6729 cfn 6730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-1st 6131 df-2nd 6132 df-1o 6407 df-er 6525 df-en 6731 df-dom 6732 df-fin 6733 df-dju 7027 df-inl 7036 df-inr 7037 df-case 7073 |
This theorem is referenced by: inffinp1 12395 |
Copyright terms: Public domain | W3C validator |