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Mirrors > Home > ILE Home > Th. List > nninff | Unicode version |
Description: An element of ℕ∞ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.) |
Ref | Expression |
---|---|
nninff | ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 5482 | . . . . . 6 | |
2 | fveq1 5482 | . . . . . 6 | |
3 | 1, 2 | sseq12d 3171 | . . . . 5 |
4 | 3 | ralbidv 2464 | . . . 4 |
5 | df-nninf 7079 | . . . 4 ℕ∞ | |
6 | 4, 5 | elrab2 2883 | . . 3 ℕ∞ |
7 | 6 | simplbi 272 | . 2 ℕ∞ |
8 | elmapi 6630 | . 2 | |
9 | 7, 8 | syl 14 | 1 ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1342 wcel 2135 wral 2442 wss 3114 csuc 4340 com 4564 wf 5181 cfv 5185 (class class class)co 5839 c2o 6372 cmap 6608 ℕ∞xnninf 7078 |
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 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 |
This theorem depends on definitions: df-bi 116 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-rab 2451 df-v 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-id 4268 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-fv 5193 df-ov 5842 df-oprab 5843 df-mpo 5844 df-map 6610 df-nninf 7079 |
This theorem is referenced by: nnnninfeq 7086 nnnninfeq2 7087 nninfisol 7091 nnsf 13778 peano4nninf 13779 nninfall 13782 nninfsellemeqinf 13789 |
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