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Mirrors > Home > ILE Home > Th. List > ennnfonelemk | Unicode version |
Description: Lemma for ennnfone 12451. (Contributed by Jim Kingdon, 15-Jul-2023.) |
Ref | Expression |
---|---|
ennnfonelemk.f |
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ennnfonelemk.k |
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ennnfonelemk.n |
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ennnfonelemk.j |
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Ref | Expression |
---|---|
ennnfonelemk |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 110 |
. 2
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2 | eqimss2 3225 |
. . . 4
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3 | 2 | adantl 277 |
. . 3
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4 | eqid 2189 |
. . . . 5
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5 | fveq2 5531 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | neeq2d 2379 |
. . . . . . . 8
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7 | ennnfonelemk.j |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 7 | adantr 276 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | simpr 110 |
. . . . . . . . . 10
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10 | ennnfonelemk.k |
. . . . . . . . . . . 12
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11 | 10 | adantr 276 |
. . . . . . . . . . 11
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12 | ennnfonelemk.n |
. . . . . . . . . . . 12
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13 | 12 | adantr 276 |
. . . . . . . . . . 11
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14 | nnsucsssuc 6512 |
. . . . . . . . . . 11
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15 | 11, 13, 14 | syl2anc 411 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 9, 15 | mpbid 147 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | peano2 4609 |
. . . . . . . . . . 11
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18 | nnord 4626 |
. . . . . . . . . . 11
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19 | 13, 17, 18 | 3syl 17 |
. . . . . . . . . 10
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20 | ordelsuc 4519 |
. . . . . . . . . 10
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21 | 11, 19, 20 | syl2anc 411 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 16, 21 | mpbird 167 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 6, 8, 22 | rspcdva 2861 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | neneqd 2381 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | ex 115 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 4, 25 | mt2i 645 |
. . . 4
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27 | 26 | adantr 276 |
. . 3
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28 | 3, 27 | pm2.21dd 621 |
. 2
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29 | 12 | adantr 276 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | nnon 4624 |
. . . . 5
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31 | 29, 30 | syl 14 |
. . . 4
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32 | simpr 110 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | onelss 4402 |
. . . 4
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34 | 31, 32, 33 | sylc 62 |
. . 3
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35 | 26 | adantr 276 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 34, 35 | pm2.21dd 621 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | nntri3or 6513 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
38 | 12, 10, 37 | syl2anc 411 |
. 2
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39 | 1, 28, 36, 38 | mpjao3dan 1318 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-tr 4117 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-iota 5193 df-fv 5240 |
This theorem is referenced by: ennnfonelemex 12440 |
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