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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 4564 | . . . . . 6 | |
2 | 1 | snid 3601 | . . . . 5 |
3 | elun2 3285 | . . . . 5 | |
4 | breq1 3979 | . . . . . 6 | |
5 | 4 | elrab3 2878 | . . . . 5 |
6 | 2, 3, 5 | mp2b 8 | . . . 4 |
7 | 6 | biimpri 132 | . . 3 |
8 | elrabi 2874 | . . . . . . 7 | |
9 | elun 3258 | . . . . . . 7 | |
10 | 8, 9 | sylib 121 | . . . . . 6 |
11 | ordom 4578 | . . . . . . . 8 | |
12 | ordelss 4351 | . . . . . . . 8 | |
13 | 11, 12 | mpan 421 | . . . . . . 7 |
14 | elsni 3588 | . . . . . . . 8 | |
15 | eqimss 3191 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | 13, 16 | jaoi 706 | . . . . . 6 |
18 | 10, 17 | syl 14 | . . . . 5 |
19 | 18 | adantl 275 | . . . 4 |
20 | 19 | ralrimiva 2537 | . . 3 |
21 | ssunieq 3816 | . . 3 | |
22 | 7, 20, 21 | syl2anc 409 | . 2 |
23 | 22 | eqcomd 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wo 698 wceq 1342 wcel 2135 wral 2442 crab 2446 cun 3109 wss 3111 csn 3570 cuni 3783 class class class wbr 3976 word 4334 com 4561 cdom 6696 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-tr 4075 df-iord 4338 df-suc 4343 df-iom 4562 |
This theorem is referenced by: hashinfom 10680 |
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