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Mirrors > Home > MPE Home > Th. List > 0bits | Structured version Visualization version GIF version |
Description: The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.) |
Ref | Expression |
---|---|
0bits | ⊢ (bits‘0) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10713 | . . . . . . 7 ⊢ 0 ∈ V | |
2 | 1 | snid 4552 | . . . . . 6 ⊢ 0 ∈ {0} |
3 | fzo01 13210 | . . . . . 6 ⊢ (0..^1) = {0} | |
4 | 2, 3 | eleqtrri 2832 | . . . . 5 ⊢ 0 ∈ (0..^1) |
5 | 2cn 11791 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
6 | exp0 13525 | . . . . . . 7 ⊢ (2 ∈ ℂ → (2↑0) = 1) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ (2↑0) = 1 |
8 | 7 | oveq2i 7181 | . . . . 5 ⊢ (0..^(2↑0)) = (0..^1) |
9 | 4, 8 | eleqtrri 2832 | . . . 4 ⊢ 0 ∈ (0..^(2↑0)) |
10 | 0z 12073 | . . . . 5 ⊢ 0 ∈ ℤ | |
11 | 0nn0 11991 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
12 | bitsfzo 15878 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℕ0) → (0 ∈ (0..^(2↑0)) ↔ (bits‘0) ⊆ (0..^0))) | |
13 | 10, 11, 12 | mp2an 692 | . . . 4 ⊢ (0 ∈ (0..^(2↑0)) ↔ (bits‘0) ⊆ (0..^0)) |
14 | 9, 13 | mpbi 233 | . . 3 ⊢ (bits‘0) ⊆ (0..^0) |
15 | fzo0 13152 | . . 3 ⊢ (0..^0) = ∅ | |
16 | 14, 15 | sseqtri 3913 | . 2 ⊢ (bits‘0) ⊆ ∅ |
17 | 0ss 4285 | . 2 ⊢ ∅ ⊆ (bits‘0) | |
18 | 16, 17 | eqssi 3893 | 1 ⊢ (bits‘0) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 ∈ wcel 2114 ⊆ wss 3843 ∅c0 4211 {csn 4516 ‘cfv 6339 (class class class)co 7170 ℂcc 10613 0cc0 10615 1c1 10616 2c2 11771 ℕ0cn0 11976 ℤcz 12062 ..^cfzo 13124 ↑cexp 13521 bitscbits 15862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-fz 12982 df-fzo 13125 df-fl 13253 df-seq 13461 df-exp 13522 df-dvds 15700 df-bits 15865 |
This theorem is referenced by: m1bits 15883 sadcadd 15901 sadadd2 15903 bitsres 15916 smumullem 15935 eulerpartgbij 31909 eulerpartlemmf 31912 eulerpartlemgvv 31913 eulerpartlemgh 31915 |
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