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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 3238 | . . . . 5 | |
2 | 1 | adantr 274 | . . . 4 |
3 | endom 6650 | . . . . 5 | |
4 | 3 | adantl 275 | . . . 4 |
5 | breq1 3927 | . . . . 5 | |
6 | 5 | elrab 2835 | . . . 4 |
7 | 2, 4, 6 | sylanbrc 413 | . . 3 |
8 | breq1 3927 | . . . . . . . . . . . 12 | |
9 | 8 | elrab 2835 | . . . . . . . . . . 11 |
10 | 9 | biimpi 119 | . . . . . . . . . 10 |
11 | 10 | adantl 275 | . . . . . . . . 9 |
12 | 11 | simprd 113 | . . . . . . . 8 |
13 | simplr 519 | . . . . . . . . 9 | |
14 | 13 | ensymd 6670 | . . . . . . . 8 |
15 | domentr 6678 | . . . . . . . 8 | |
16 | 12, 14, 15 | syl2anc 408 | . . . . . . 7 |
17 | 16 | adantr 274 | . . . . . 6 |
18 | simpr 109 | . . . . . . 7 | |
19 | simplll 522 | . . . . . . 7 | |
20 | nndomo 6751 | . . . . . . 7 | |
21 | 18, 19, 20 | syl2anc 408 | . . . . . 6 |
22 | 17, 21 | mpbid 146 | . . . . 5 |
23 | nnfi 6759 | . . . . . . . 8 | |
24 | 23 | ad3antrrr 483 | . . . . . . 7 |
25 | 14 | adantr 274 | . . . . . . 7 |
26 | enfii 6761 | . . . . . . 7 | |
27 | 24, 25, 26 | syl2anc 408 | . . . . . 6 |
28 | 12 | adantr 274 | . . . . . . . 8 |
29 | elsni 3540 | . . . . . . . . . 10 | |
30 | 29 | breq1d 3934 | . . . . . . . . 9 |
31 | 30 | adantl 275 | . . . . . . . 8 |
32 | 28, 31 | mpbid 146 | . . . . . . 7 |
33 | infnfi 6782 | . . . . . . 7 | |
34 | 32, 33 | syl 14 | . . . . . 6 |
35 | 27, 34 | pm2.21dd 609 | . . . . 5 |
36 | 11 | simpld 111 | . . . . . 6 |
37 | elun 3212 | . . . . . 6 | |
38 | 36, 37 | sylib 121 | . . . . 5 |
39 | 22, 35, 38 | mpjaodan 787 | . . . 4 |
40 | 39 | ralrimiva 2503 | . . 3 |
41 | ssunieq 3764 | . . 3 | |
42 | 7, 40, 41 | syl2anc 408 | . 2 |
43 | 42 | eqcomd 2143 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 wral 2414 crab 2418 cun 3064 wss 3066 csn 3522 cuni 3731 class class class wbr 3924 com 4499 cen 6625 cdom 6626 cfn 6627 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 |
This theorem is referenced by: hashennn 10519 |
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