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Mirrors > Home > ILE Home > Th. List > nninfisollemne | Unicode version |
Description: Lemma for nninfisol 7088. A case where is a successor and and are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.) |
Ref | Expression |
---|---|
nninfisol.x | ℕ∞ |
nninfisol.0 | |
nninfisol.n | |
nninfisollemne.s | |
nninfisollemne.0 |
Ref | Expression |
---|---|
nninfisollemne | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nninfisollemne.0 | . . . . 5 | |
2 | 1 | adantr 274 | . . . 4 |
3 | simpr 109 | . . . . . . . 8 | |
4 | 3 | fveq1d 5482 | . . . . . . 7 |
5 | eqid 2164 | . . . . . . . . . 10 | |
6 | eleq1 2227 | . . . . . . . . . . 11 | |
7 | 6 | ifbid 3536 | . . . . . . . . . 10 |
8 | nninfisol.n | . . . . . . . . . . 11 | |
9 | nnpredcl 4594 | . . . . . . . . . . 11 | |
10 | 8, 9 | syl 14 | . . . . . . . . . 10 |
11 | nninfisollemne.s | . . . . . . . . . . . . 13 | |
12 | nnpredlt 4595 | . . . . . . . . . . . . 13 | |
13 | 8, 11, 12 | syl2anc 409 | . . . . . . . . . . . 12 |
14 | 13 | iftrued 3522 | . . . . . . . . . . 11 |
15 | 1lt2o 6401 | . . . . . . . . . . 11 | |
16 | 14, 15 | eqeltrdi 2255 | . . . . . . . . . 10 |
17 | 5, 7, 10, 16 | fvmptd3 5573 | . . . . . . . . 9 |
18 | 17, 14 | eqtrd 2197 | . . . . . . . 8 |
19 | 18 | adantr 274 | . . . . . . 7 |
20 | 4, 19 | eqtr3d 2199 | . . . . . 6 |
21 | 1n0 6391 | . . . . . 6 | |
22 | pm13.181 2416 | . . . . . 6 | |
23 | 20, 21, 22 | sylancl 410 | . . . . 5 |
24 | 23 | neneqd 2355 | . . . 4 |
25 | 2, 24 | pm2.65da 651 | . . 3 |
26 | 25 | olcd 724 | . 2 |
27 | df-dc 825 | . 2 DECID | |
28 | 26, 27 | sylibr 133 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 824 wceq 1342 wcel 2135 wne 2334 c0 3404 cif 3515 cuni 3783 cmpt 4037 com 4561 cfv 5182 c1o 6368 c2o 6369 ℕ∞xnninf 7075 |
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-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 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-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 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-mpt 4039 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-iota 5147 df-fun 5184 df-fv 5190 df-1o 6375 df-2o 6376 |
This theorem is referenced by: nninfisol 7088 |
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