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Mirrors > Home > ILE Home > Th. List > nninfisollemne | Unicode version |
Description: Lemma for nninfisol 7077. 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 5471 | . . . . . . 7 |
5 | eqid 2157 | . . . . . . . . . 10 | |
6 | eleq1 2220 | . . . . . . . . . . 11 | |
7 | 6 | ifbid 3526 | . . . . . . . . . 10 |
8 | nninfisol.n | . . . . . . . . . . 11 | |
9 | nnpredcl 4583 | . . . . . . . . . . 11 | |
10 | 8, 9 | syl 14 | . . . . . . . . . 10 |
11 | nninfisollemne.s | . . . . . . . . . . . . 13 | |
12 | nnpredlt 4584 | . . . . . . . . . . . . 13 | |
13 | 8, 11, 12 | syl2anc 409 | . . . . . . . . . . . 12 |
14 | 13 | iftrued 3512 | . . . . . . . . . . 11 |
15 | 1lt2o 6390 | . . . . . . . . . . 11 | |
16 | 14, 15 | eqeltrdi 2248 | . . . . . . . . . 10 |
17 | 5, 7, 10, 16 | fvmptd3 5562 | . . . . . . . . 9 |
18 | 17, 14 | eqtrd 2190 | . . . . . . . 8 |
19 | 18 | adantr 274 | . . . . . . 7 |
20 | 4, 19 | eqtr3d 2192 | . . . . . 6 |
21 | 1n0 6380 | . . . . . 6 | |
22 | pm13.181 2409 | . . . . . 6 | |
23 | 20, 21, 22 | sylancl 410 | . . . . 5 |
24 | 23 | neneqd 2348 | . . . 4 |
25 | 2, 24 | pm2.65da 651 | . . 3 |
26 | 25 | olcd 724 | . 2 |
27 | df-dc 821 | . 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 820 wceq 1335 wcel 2128 wne 2327 c0 3394 cif 3505 cuni 3773 cmpt 4026 com 4550 cfv 5171 c1o 6357 c2o 6358 ℕ∞xnninf 7064 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-1o 6364 df-2o 6365 |
This theorem is referenced by: nninfisol 7077 |
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