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Mirrors > Home > ILE Home > Th. List > hashinfuni | Unicode version |
Description: The ordinal size of an infinite set is . (Contributed by Jim Kingdon, 20-Feb-2022.) |
Ref | Expression |
---|---|
hashinfuni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 4477 | . . . . . 6 | |
2 | 1 | snid 3526 | . . . . 5 |
3 | elun2 3214 | . . . . 5 | |
4 | breq1 3902 | . . . . . 6 | |
5 | 4 | elrab3 2814 | . . . . 5 |
6 | 2, 3, 5 | mp2b 8 | . . . 4 |
7 | 6 | biimpri 132 | . . 3 |
8 | elrabi 2810 | . . . . . . 7 | |
9 | elun 3187 | . . . . . . 7 | |
10 | 8, 9 | sylib 121 | . . . . . 6 |
11 | ordom 4490 | . . . . . . . 8 | |
12 | ordelss 4271 | . . . . . . . 8 | |
13 | 11, 12 | mpan 420 | . . . . . . 7 |
14 | elsni 3515 | . . . . . . . 8 | |
15 | eqimss 3121 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | 13, 16 | jaoi 690 | . . . . . 6 |
18 | 10, 17 | syl 14 | . . . . 5 |
19 | 18 | adantl 275 | . . . 4 |
20 | 19 | ralrimiva 2482 | . . 3 |
21 | ssunieq 3739 | . . 3 | |
22 | 7, 20, 21 | syl2anc 408 | . 2 |
23 | 22 | eqcomd 2123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wo 682 wceq 1316 wcel 1465 wral 2393 crab 2397 cun 3039 wss 3041 csn 3497 cuni 3706 class class class wbr 3899 word 4254 com 4474 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-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-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 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-br 3900 df-tr 3997 df-iord 4258 df-suc 4263 df-iom 4475 |
This theorem is referenced by: hashinfom 10492 |
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