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Mirrors > Home > MPE Home > Th. List > fin23lem41 | Unicode version |
Description: Lemma for fin23 8790. A set which satisfies the descending sequence condition must be III-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
fin23lem40.f |
Ref | Expression |
---|---|
fin23lem41 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 7547 | . . . . 5 | |
2 | fin23lem40.f | . . . . . . . . . 10 | |
3 | 2 | fin23lem33 8746 | . . . . . . . . 9 |
4 | 3 | adantl 466 | . . . . . . . 8 |
5 | ssv 3523 | . . . . . . . . . . 11 | |
6 | f1ss 5791 | . . . . . . . . . . 11 | |
7 | 5, 6 | mpan2 671 | . . . . . . . . . 10 |
8 | 7 | ad2antrr 725 | . . . . . . . . 9 |
9 | f1f 5786 | . . . . . . . . . . . 12 | |
10 | frn 5742 | . . . . . . . . . . . 12 | |
11 | uniss 4270 | . . . . . . . . . . . 12 | |
12 | 9, 10, 11 | 3syl 20 | . . . . . . . . . . 11 |
13 | unipw 4702 | . . . . . . . . . . 11 | |
14 | 12, 13 | syl6sseq 3549 | . . . . . . . . . 10 |
15 | 14 | ad2antrr 725 | . . . . . . . . 9 |
16 | f1eq1 5781 | . . . . . . . . . . . . . 14 | |
17 | rneq 5233 | . . . . . . . . . . . . . . . 16 | |
18 | 17 | unieqd 4259 | . . . . . . . . . . . . . . 15 |
19 | 18 | sseq1d 3530 | . . . . . . . . . . . . . 14 |
20 | 16, 19 | anbi12d 710 | . . . . . . . . . . . . 13 |
21 | fveq2 5871 | . . . . . . . . . . . . . . 15 | |
22 | f1eq1 5781 | . . . . . . . . . . . . . . 15 | |
23 | 21, 22 | syl 16 | . . . . . . . . . . . . . 14 |
24 | 21 | rneqd 5235 | . . . . . . . . . . . . . . . 16 |
25 | 24 | unieqd 4259 | . . . . . . . . . . . . . . 15 |
26 | 25, 18 | psseq12d 3597 | . . . . . . . . . . . . . 14 |
27 | 23, 26 | anbi12d 710 | . . . . . . . . . . . . 13 |
28 | 20, 27 | imbi12d 320 | . . . . . . . . . . . 12 |
29 | 28 | cbvalv 2023 | . . . . . . . . . . 11 |
30 | 29 | biimpi 194 | . . . . . . . . . 10 |
31 | 30 | adantl 466 | . . . . . . . . 9 |
32 | eqid 2457 | . . . . . . . . 9 | |
33 | 2, 8, 15, 31, 32 | fin23lem39 8751 | . . . . . . . 8 |
34 | 4, 33 | exlimddv 1726 | . . . . . . 7 |
35 | 34 | pm2.01da 442 | . . . . . 6 |
36 | 35 | exlimiv 1722 | . . . . 5 |
37 | 1, 36 | syl 16 | . . . 4 |
38 | 37 | con2i 120 | . . 3 |
39 | pwexg 4636 | . . . 4 | |
40 | isfin4-2 8715 | . . . 4 | |
41 | 39, 40 | syl 16 | . . 3 |
42 | 38, 41 | mpbird 232 | . 2 |
43 | isfin3 8697 | . 2 | |
44 | 42, 43 | sylibr 212 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -. wn 3 -> wi 4
<-> wb 184 /\ wa 369 A. wal 1393
= wceq 1395 E. wex 1612 e. wcel 1818
{ cab 2442 A. wral 2807 cvv 3109
C_ wss 3475 C. wpss 3476 ~P cpw 4012
U. cuni 4249 |^| cint 4286 class class class wbr 4452
suc csuc 4885
ran crn 5005 |` cres 5006 --> wf 5589
-1-1-> wf1 5590
` cfv 5593 (class class class)co 6296
com 6700
rec crdg 7094
cmap 7439
cdom 7534 cfin4 8681 cfin3 8682 |
This theorem is referenced by: isf33lem 8767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-rep 4563 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-int 4287 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-se 4844 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-isom 5602 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-1st 6800 df-2nd 6801 df-recs 7061 df-rdg 7095 df-seqom 7132 df-1o 7149 df-oadd 7153 df-er 7330 df-map 7441 df-en 7537 df-dom 7538 df-sdom 7539 df-fin 7540 df-card 8341 df-fin4 8688 df-fin3 8689 |
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