![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nthruz | Structured version Visualization version GIF version |
Description: The sequence ℕ, ℕ0, and ℤ forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to ℕ0 but not ℕ and minus one belongs to ℤ but not ℕ0. This theorem refines the chain of proper subsets nthruc 16200. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nthruz | ⊢ (ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssnn0 12476 | . . 3 ⊢ ℕ ⊆ ℕ0 | |
2 | 0nn0 12488 | . . . 4 ⊢ 0 ∈ ℕ0 | |
3 | 0nnn 12249 | . . . 4 ⊢ ¬ 0 ∈ ℕ | |
4 | 2, 3 | pm3.2i 470 | . . 3 ⊢ (0 ∈ ℕ0 ∧ ¬ 0 ∈ ℕ) |
5 | ssnelpss 4106 | . . 3 ⊢ (ℕ ⊆ ℕ0 → ((0 ∈ ℕ0 ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℕ0)) | |
6 | 1, 4, 5 | mp2 9 | . 2 ⊢ ℕ ⊊ ℕ0 |
7 | nn0ssz 12582 | . . 3 ⊢ ℕ0 ⊆ ℤ | |
8 | neg1z 12599 | . . . 4 ⊢ -1 ∈ ℤ | |
9 | neg1lt0 12330 | . . . . 5 ⊢ -1 < 0 | |
10 | nn0nlt0 12499 | . . . . 5 ⊢ (-1 ∈ ℕ0 → ¬ -1 < 0) | |
11 | 9, 10 | mt2 199 | . . . 4 ⊢ ¬ -1 ∈ ℕ0 |
12 | 8, 11 | pm3.2i 470 | . . 3 ⊢ (-1 ∈ ℤ ∧ ¬ -1 ∈ ℕ0) |
13 | ssnelpss 4106 | . . 3 ⊢ (ℕ0 ⊆ ℤ → ((-1 ∈ ℤ ∧ ¬ -1 ∈ ℕ0) → ℕ0 ⊊ ℤ)) | |
14 | 7, 12, 13 | mp2 9 | . 2 ⊢ ℕ0 ⊊ ℤ |
15 | 6, 14 | pm3.2i 470 | 1 ⊢ (ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∈ wcel 2098 ⊆ wss 3943 ⊊ wpss 3944 class class class wbr 5141 0cc0 11109 1c1 11110 < clt 11249 -cneg 11446 ℕcn 12213 ℕ0cn0 12473 ℤcz 12559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |