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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 4507 | . . . . . 6 | |
2 | 1 | snid 3556 | . . . . 5 |
3 | elun2 3244 | . . . . 5 | |
4 | breq1 3932 | . . . . . 6 | |
5 | 4 | elrab3 2841 | . . . . 5 |
6 | 2, 3, 5 | mp2b 8 | . . . 4 |
7 | 6 | biimpri 132 | . . 3 |
8 | elrabi 2837 | . . . . . . 7 | |
9 | elun 3217 | . . . . . . 7 | |
10 | 8, 9 | sylib 121 | . . . . . 6 |
11 | ordom 4520 | . . . . . . . 8 | |
12 | ordelss 4301 | . . . . . . . 8 | |
13 | 11, 12 | mpan 420 | . . . . . . 7 |
14 | elsni 3545 | . . . . . . . 8 | |
15 | eqimss 3151 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | 13, 16 | jaoi 705 | . . . . . 6 |
18 | 10, 17 | syl 14 | . . . . 5 |
19 | 18 | adantl 275 | . . . 4 |
20 | 19 | ralrimiva 2505 | . . 3 |
21 | ssunieq 3769 | . . 3 | |
22 | 7, 20, 21 | syl2anc 408 | . 2 |
23 | 22 | eqcomd 2145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wo 697 wceq 1331 wcel 1480 wral 2416 crab 2420 cun 3069 wss 3071 csn 3527 cuni 3736 class class class wbr 3929 word 4284 com 4504 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-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-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 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-br 3930 df-tr 4027 df-iord 4288 df-suc 4293 df-iom 4505 |
This theorem is referenced by: hashinfom 10524 |
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