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Mirrors > Home > ILE Home > Th. List > nninfisollem0 | Unicode version |
Description: Lemma for nninfisol 7109. The case where is zero. (Contributed by Jim Kingdon, 13-Sep-2024.) |
Ref | Expression |
---|---|
nninfisol.x | ℕ∞ |
nninfisol.0 | |
nninfisol.n | |
nninfisollem0.0 |
Ref | Expression |
---|---|
nninfisollem0 | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nninfisol.x | . . . . 5 ℕ∞ | |
2 | nninfisol.n | . . . . 5 | |
3 | ral0 3516 | . . . . . 6 | |
4 | nninfisollem0.0 | . . . . . . 7 | |
5 | 4 | raleqdv 2671 | . . . . . 6 |
6 | 3, 5 | mpbiri 167 | . . . . 5 |
7 | nninfisol.0 | . . . . 5 | |
8 | 1, 2, 6, 7 | nnnninfeq 7104 | . . . 4 |
9 | 8 | eqcomd 2176 | . . 3 |
10 | 9 | orcd 728 | . 2 |
11 | df-dc 830 | . 2 DECID | |
12 | 10, 11 | sylibr 133 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 703 DECID wdc 829 wceq 1348 wcel 2141 wral 2448 c0 3414 cif 3526 cmpt 4050 com 4574 cfv 5198 c1o 6388 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-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 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-rab 2457 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-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1o 6395 df-2o 6396 df-map 6628 df-nninf 7097 |
This theorem is referenced by: nninfisol 7109 |
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