Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3284 | . . . . 5 | |
2 | 1 | adantr 274 | . . . 4 |
3 | endom 6720 | . . . . 5 | |
4 | 3 | adantl 275 | . . . 4 |
5 | breq1 3979 | . . . . 5 | |
6 | 5 | elrab 2877 | . . . 4 |
7 | 2, 4, 6 | sylanbrc 414 | . . 3 |
8 | breq1 3979 | . . . . . . . . . . . 12 | |
9 | 8 | elrab 2877 | . . . . . . . . . . 11 |
10 | 9 | biimpi 119 | . . . . . . . . . 10 |
11 | 10 | adantl 275 | . . . . . . . . 9 |
12 | 11 | simprd 113 | . . . . . . . 8 |
13 | simplr 520 | . . . . . . . . 9 | |
14 | 13 | ensymd 6740 | . . . . . . . 8 |
15 | domentr 6748 | . . . . . . . 8 | |
16 | 12, 14, 15 | syl2anc 409 | . . . . . . 7 |
17 | 16 | adantr 274 | . . . . . 6 |
18 | simpr 109 | . . . . . . 7 | |
19 | simplll 523 | . . . . . . 7 | |
20 | nndomo 6821 | . . . . . . 7 | |
21 | 18, 19, 20 | syl2anc 409 | . . . . . 6 |
22 | 17, 21 | mpbid 146 | . . . . 5 |
23 | nnfi 6829 | . . . . . . . 8 | |
24 | 23 | ad3antrrr 484 | . . . . . . 7 |
25 | 14 | adantr 274 | . . . . . . 7 |
26 | enfii 6831 | . . . . . . 7 | |
27 | 24, 25, 26 | syl2anc 409 | . . . . . 6 |
28 | 12 | adantr 274 | . . . . . . . 8 |
29 | elsni 3588 | . . . . . . . . . 10 | |
30 | 29 | breq1d 3986 | . . . . . . . . 9 |
31 | 30 | adantl 275 | . . . . . . . 8 |
32 | 28, 31 | mpbid 146 | . . . . . . 7 |
33 | infnfi 6852 | . . . . . . 7 | |
34 | 32, 33 | syl 14 | . . . . . 6 |
35 | 27, 34 | pm2.21dd 610 | . . . . 5 |
36 | 11 | simpld 111 | . . . . . 6 |
37 | elun 3258 | . . . . . 6 | |
38 | 36, 37 | sylib 121 | . . . . 5 |
39 | 22, 35, 38 | mpjaodan 788 | . . . 4 |
40 | 39 | ralrimiva 2537 | . . 3 |
41 | ssunieq 3816 | . . 3 | |
42 | 7, 40, 41 | syl2anc 409 | . 2 |
43 | 42 | eqcomd 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 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 com 4561 cen 6695 cdom 6696 cfn 6697 |
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-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 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-opab 4038 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 |
This theorem is referenced by: hashennn 10682 |
Copyright terms: Public domain | W3C validator |