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Mirrors > Home > ILE Home > Th. List > nninfisollemeq | Unicode version |
Description: Lemma for nninfisol 7077. The case where is a successor and and are equal. (Contributed by Jim Kingdon, 13-Sep-2024.) |
Ref | Expression |
---|---|
nninfisol.x | ℕ∞ |
nninfisol.0 | |
nninfisol.n | |
nninfisollemeq.s | |
nninfisollemeq.0 |
Ref | Expression |
---|---|
nninfisollemeq | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nninfisol.x | . . . . 5 ℕ∞ | |
2 | nninfisol.n | . . . . 5 | |
3 | nninfisollemeq.0 | . . . . 5 | |
4 | nninfisol.0 | . . . . 5 | |
5 | 1, 2, 3, 4 | nnnninfeq2 7073 | . . . 4 |
6 | 5 | eqcomd 2163 | . . 3 |
7 | 6 | orcd 723 | . 2 |
8 | df-dc 821 | . 2 DECID | |
9 | 7, 8 | sylibr 133 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 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 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-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 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-rab 2444 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-rn 4598 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1o 6364 df-2o 6365 df-map 6596 df-nninf 7065 |
This theorem is referenced by: nninfisol 7077 |
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