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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 6643 | . . . 4 | |
2 | 1 | adantr 274 | . . 3 |
3 | vex 2689 | . . . . 5 | |
4 | imaexg 4893 | . . . . 5 | |
5 | 3, 4 | ax-mp 5 | . . . 4 |
6 | imassrn 4892 | . . . . . 6 | |
7 | simpr 109 | . . . . . . 7 | |
8 | f1f 5328 | . . . . . . 7 | |
9 | frn 5281 | . . . . . . 7 | |
10 | 7, 8, 9 | 3syl 17 | . . . . . 6 |
11 | 6, 10 | sstrid 3108 | . . . . 5 |
12 | ordom 4520 | . . . . . . . 8 | |
13 | ordelss 4301 | . . . . . . . 8 | |
14 | 12, 13 | mpan 420 | . . . . . . 7 |
15 | 14 | ad2antlr 480 | . . . . . 6 |
16 | simplr 519 | . . . . . 6 | |
17 | f1imaeng 6686 | . . . . . 6 | |
18 | 7, 15, 16, 17 | syl3anc 1216 | . . . . 5 |
19 | 11, 18 | jca 304 | . . . 4 |
20 | sseq1 3120 | . . . . . 6 | |
21 | breq1 3932 | . . . . . 6 | |
22 | 20, 21 | anbi12d 464 | . . . . 5 |
23 | 22 | spcegv 2774 | . . . 4 |
24 | 5, 19, 23 | mpsyl 65 | . . 3 |
25 | 2, 24 | exlimddv 1870 | . 2 |
26 | 25 | ralrimiva 2505 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 wral 2416 cvv 2686 wss 3071 class class class wbr 3929 word 4284 com 4504 crn 4540 cima 4542 wf 5119 wf1 5120 cen 6632 cdom 6633 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-er 6429 df-en 6635 df-dom 6636 |
This theorem is referenced by: (None) |
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