Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > isinfinf | Unicode version |
Description: An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.) |
Ref | Expression |
---|---|
isinfinf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 6611 | . . . 4 | |
2 | 1 | adantr 274 | . . 3 |
3 | vex 2663 | . . . . 5 | |
4 | imaexg 4863 | . . . . 5 | |
5 | 3, 4 | ax-mp 5 | . . . 4 |
6 | imassrn 4862 | . . . . . 6 | |
7 | simpr 109 | . . . . . . 7 | |
8 | f1f 5298 | . . . . . . 7 | |
9 | frn 5251 | . . . . . . 7 | |
10 | 7, 8, 9 | 3syl 17 | . . . . . 6 |
11 | 6, 10 | sstrid 3078 | . . . . 5 |
12 | ordom 4490 | . . . . . . . 8 | |
13 | ordelss 4271 | . . . . . . . 8 | |
14 | 12, 13 | mpan 420 | . . . . . . 7 |
15 | 14 | ad2antlr 480 | . . . . . 6 |
16 | simplr 504 | . . . . . 6 | |
17 | f1imaeng 6654 | . . . . . 6 | |
18 | 7, 15, 16, 17 | syl3anc 1201 | . . . . 5 |
19 | 11, 18 | jca 304 | . . . 4 |
20 | sseq1 3090 | . . . . . 6 | |
21 | breq1 3902 | . . . . . 6 | |
22 | 20, 21 | anbi12d 464 | . . . . 5 |
23 | 22 | spcegv 2748 | . . . 4 |
24 | 5, 19, 23 | mpsyl 65 | . . 3 |
25 | 2, 24 | exlimddv 1854 | . 2 |
26 | 25 | ralrimiva 2482 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wex 1453 wcel 1465 wral 2393 cvv 2660 wss 3041 class class class wbr 3899 word 4254 com 4474 crn 4510 cima 4512 wf 5089 wf1 5090 cen 6600 cdom 6601 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-er 6397 df-en 6603 df-dom 6604 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |