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Mirrors > Home > ILE Home > Th. List > isnumi | Unicode version |
Description: A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
isnumi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3902 | . . . . 5 | |
2 | 1 | rspcev 2763 | . . . 4 |
3 | intexrabim 4048 | . . . 4 | |
4 | 2, 3 | syl 14 | . . 3 |
5 | encv 6608 | . . . . . 6 | |
6 | 5 | simprd 113 | . . . . 5 |
7 | breq2 3903 | . . . . . . . . 9 | |
8 | 7 | rabbidv 2649 | . . . . . . . 8 |
9 | 8 | inteqd 3746 | . . . . . . 7 |
10 | 9 | eleq1d 2186 | . . . . . 6 |
11 | 10 | elrab3 2814 | . . . . 5 |
12 | 6, 11 | syl 14 | . . . 4 |
13 | 12 | adantl 275 | . . 3 |
14 | 4, 13 | mpbird 166 | . 2 |
15 | df-card 7004 | . . 3 | |
16 | 15 | dmmpt 5004 | . 2 |
17 | 14, 16 | eleqtrrdi 2211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wrex 2394 crab 2397 cvv 2660 cint 3741 class class class wbr 3899 con0 4255 cdm 4509 cen 6600 ccrd 7003 |
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-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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
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-rab 2402 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-xp 4515 df-rel 4516 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-en 6603 df-card 7004 |
This theorem is referenced by: finnum 7007 onenon 7008 |
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