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Mirrors > Home > ILE Home > Th. List > hashennnuni | Unicode version |
Description: The ordinal size of a set equinumerous to an element of is that element of . (Contributed by Jim Kingdon, 20-Feb-2022.) |
Ref | Expression |
---|---|
hashennnuni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun1 3213 | . . . . 5 | |
2 | 1 | adantr 274 | . . . 4 |
3 | endom 6625 | . . . . 5 | |
4 | 3 | adantl 275 | . . . 4 |
5 | breq1 3902 | . . . . 5 | |
6 | 5 | elrab 2813 | . . . 4 |
7 | 2, 4, 6 | sylanbrc 413 | . . 3 |
8 | breq1 3902 | . . . . . . . . . . . 12 | |
9 | 8 | elrab 2813 | . . . . . . . . . . 11 |
10 | 9 | biimpi 119 | . . . . . . . . . 10 |
11 | 10 | adantl 275 | . . . . . . . . 9 |
12 | 11 | simprd 113 | . . . . . . . 8 |
13 | simplr 504 | . . . . . . . . 9 | |
14 | 13 | ensymd 6645 | . . . . . . . 8 |
15 | domentr 6653 | . . . . . . . 8 | |
16 | 12, 14, 15 | syl2anc 408 | . . . . . . 7 |
17 | 16 | adantr 274 | . . . . . 6 |
18 | simpr 109 | . . . . . . 7 | |
19 | simplll 507 | . . . . . . 7 | |
20 | nndomo 6726 | . . . . . . 7 | |
21 | 18, 19, 20 | syl2anc 408 | . . . . . 6 |
22 | 17, 21 | mpbid 146 | . . . . 5 |
23 | nnfi 6734 | . . . . . . . 8 | |
24 | 23 | ad3antrrr 483 | . . . . . . 7 |
25 | 14 | adantr 274 | . . . . . . 7 |
26 | enfii 6736 | . . . . . . 7 | |
27 | 24, 25, 26 | syl2anc 408 | . . . . . 6 |
28 | 12 | adantr 274 | . . . . . . . 8 |
29 | elsni 3515 | . . . . . . . . . 10 | |
30 | 29 | breq1d 3909 | . . . . . . . . 9 |
31 | 30 | adantl 275 | . . . . . . . 8 |
32 | 28, 31 | mpbid 146 | . . . . . . 7 |
33 | infnfi 6757 | . . . . . . 7 | |
34 | 32, 33 | syl 14 | . . . . . 6 |
35 | 27, 34 | pm2.21dd 594 | . . . . 5 |
36 | 11 | simpld 111 | . . . . . 6 |
37 | elun 3187 | . . . . . 6 | |
38 | 36, 37 | sylib 121 | . . . . 5 |
39 | 22, 35, 38 | mpjaodan 772 | . . . 4 |
40 | 39 | ralrimiva 2482 | . . 3 |
41 | ssunieq 3739 | . . 3 | |
42 | 7, 40, 41 | syl2anc 408 | . 2 |
43 | 42 | eqcomd 2123 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 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 com 4474 cen 6600 cdom 6601 cfn 6602 |
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-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 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-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 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-opab 3960 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-er 6397 df-en 6603 df-dom 6604 df-fin 6605 |
This theorem is referenced by: hashennn 10494 |
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