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Mirrors > Home > MPE Home > Th. List > nnssz | Structured version Visualization version GIF version |
Description: Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022.) |
Ref | Expression |
---|---|
nnssz | ⊢ ℕ ⊆ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 11645 | . . 3 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ) | |
2 | 3mix2 1327 | . . 3 ⊢ (𝑥 ∈ ℕ → (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ)) | |
3 | elz 11984 | . . 3 ⊢ (𝑥 ∈ ℤ ↔ (𝑥 ∈ ℝ ∧ (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ))) | |
4 | 1, 2, 3 | sylanbrc 585 | . 2 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ) |
5 | 4 | ssriv 3971 | 1 ⊢ ℕ ⊆ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ℝcr 10536 0cc0 10537 -cneg 10871 ℕcn 11638 ℤcz 11982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-i2m1 10605 ax-1ne0 10606 ax-rrecex 10609 ax-cnre 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-neg 10873 df-nn 11639 df-z 11983 |
This theorem is referenced by: nn0ssz 12004 nnz 12005 nnzi 12007 zmin 12345 nnssq 12358 divcnvshft 15210 znnen 15565 nthruc 15605 alzdvds 15670 evennn2n 15700 lcmfnnval 15968 lcmfnncl 15973 pclem 16175 prmreclem1 16252 ftalem5 25654 2sqreunnltblem 26027 archiabllem1b 30821 reprsuc 31886 divcnvlin 32964 dffltz 39291 diophin 39389 hashnzfzclim 40674 |
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