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Mirrors > Home > ILE Home > Th. List > ennnfonelemk | Unicode version |
Description: Lemma for ennnfone 11938. (Contributed by Jim Kingdon, 15-Jul-2023.) |
Ref | Expression |
---|---|
ennnfonelemk.f | |
ennnfonelemk.k | |
ennnfonelemk.n | |
ennnfonelemk.j |
Ref | Expression |
---|---|
ennnfonelemk |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . 2 | |
2 | eqimss2 3152 | . . . 4 | |
3 | 2 | adantl 275 | . . 3 |
4 | eqid 2139 | . . . . 5 | |
5 | fveq2 5421 | . . . . . . . . 9 | |
6 | 5 | neeq2d 2327 | . . . . . . . 8 |
7 | ennnfonelemk.j | . . . . . . . . 9 | |
8 | 7 | adantr 274 | . . . . . . . 8 |
9 | simpr 109 | . . . . . . . . . 10 | |
10 | ennnfonelemk.k | . . . . . . . . . . . 12 | |
11 | 10 | adantr 274 | . . . . . . . . . . 11 |
12 | ennnfonelemk.n | . . . . . . . . . . . 12 | |
13 | 12 | adantr 274 | . . . . . . . . . . 11 |
14 | nnsucsssuc 6388 | . . . . . . . . . . 11 | |
15 | 11, 13, 14 | syl2anc 408 | . . . . . . . . . 10 |
16 | 9, 15 | mpbid 146 | . . . . . . . . 9 |
17 | peano2 4509 | . . . . . . . . . . 11 | |
18 | nnord 4525 | . . . . . . . . . . 11 | |
19 | 13, 17, 18 | 3syl 17 | . . . . . . . . . 10 |
20 | ordelsuc 4421 | . . . . . . . . . 10 | |
21 | 11, 19, 20 | syl2anc 408 | . . . . . . . . 9 |
22 | 16, 21 | mpbird 166 | . . . . . . . 8 |
23 | 6, 8, 22 | rspcdva 2794 | . . . . . . 7 |
24 | 23 | neneqd 2329 | . . . . . 6 |
25 | 24 | ex 114 | . . . . 5 |
26 | 4, 25 | mt2i 633 | . . . 4 |
27 | 26 | adantr 274 | . . 3 |
28 | 3, 27 | pm2.21dd 609 | . 2 |
29 | 12 | adantr 274 | . . . . 5 |
30 | nnon 4523 | . . . . 5 | |
31 | 29, 30 | syl 14 | . . . 4 |
32 | simpr 109 | . . . 4 | |
33 | onelss 4309 | . . . 4 | |
34 | 31, 32, 33 | sylc 62 | . . 3 |
35 | 26 | adantr 274 | . . 3 |
36 | 34, 35 | pm2.21dd 609 | . 2 |
37 | nntri3or 6389 | . . 3 | |
38 | 12, 10, 37 | syl2anc 408 | . 2 |
39 | 1, 28, 36, 38 | mpjao3dan 1285 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 961 wceq 1331 wcel 1480 wne 2308 wral 2416 wss 3071 word 4284 con0 4285 csuc 4287 com 4504 wfo 5121 cfv 5123 |
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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-iota 5088 df-fv 5131 |
This theorem is referenced by: ennnfonelemex 11927 |
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