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Mirrors > Home > ILE Home > Th. List > nninfisollemne | Unicode version |
Description: Lemma for nninfisol 7109. 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 5498 | . . . . . . 7 |
5 | eqid 2170 | . . . . . . . . . 10 | |
6 | eleq1 2233 | . . . . . . . . . . 11 | |
7 | 6 | ifbid 3547 | . . . . . . . . . 10 |
8 | nninfisol.n | . . . . . . . . . . 11 | |
9 | nnpredcl 4607 | . . . . . . . . . . 11 | |
10 | 8, 9 | syl 14 | . . . . . . . . . 10 |
11 | nninfisollemne.s | . . . . . . . . . . . . 13 | |
12 | nnpredlt 4608 | . . . . . . . . . . . . 13 | |
13 | 8, 11, 12 | syl2anc 409 | . . . . . . . . . . . 12 |
14 | 13 | iftrued 3533 | . . . . . . . . . . 11 |
15 | 1lt2o 6421 | . . . . . . . . . . 11 | |
16 | 14, 15 | eqeltrdi 2261 | . . . . . . . . . 10 |
17 | 5, 7, 10, 16 | fvmptd3 5589 | . . . . . . . . 9 |
18 | 17, 14 | eqtrd 2203 | . . . . . . . 8 |
19 | 18 | adantr 274 | . . . . . . 7 |
20 | 4, 19 | eqtr3d 2205 | . . . . . 6 |
21 | 1n0 6411 | . . . . . 6 | |
22 | pm13.181 2422 | . . . . . 6 | |
23 | 20, 21, 22 | sylancl 411 | . . . . 5 |
24 | 23 | neneqd 2361 | . . . 4 |
25 | 2, 24 | pm2.65da 656 | . . 3 |
26 | 25 | olcd 729 | . 2 |
27 | df-dc 830 | . 2 DECID | |
28 | 26, 27 | sylibr 133 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 DECID wdc 829 wceq 1348 wcel 2141 wne 2340 c0 3414 cif 3526 cuni 3796 cmpt 4050 com 4574 cfv 5198 c1o 6388 c2o 6389 ℕ∞xnninf 7096 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-1o 6395 df-2o 6396 |
This theorem is referenced by: nninfisol 7109 |
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