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Mirrors > Home > MPE Home > Th. List > nnsscn | Structured version Visualization version GIF version |
Description: The positive integers are a subset of the complex numbers. Remark: this could also be proven from nnssre 11641 and ax-resscn 10593 at the cost of using more axioms. (Contributed by NM, 2-Aug-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
Ref | Expression |
---|---|
nnsscn | ⊢ ℕ ⊆ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10594 | . 2 ⊢ 1 ∈ ℂ | |
2 | peano2cn 10811 | . . 3 ⊢ (𝑥 ∈ ℂ → (𝑥 + 1) ∈ ℂ) | |
3 | 2 | rgen 3148 | . 2 ⊢ ∀𝑥 ∈ ℂ (𝑥 + 1) ∈ ℂ |
4 | peano5nni 11640 | . 2 ⊢ ((1 ∈ ℂ ∧ ∀𝑥 ∈ ℂ (𝑥 + 1) ∈ ℂ) → ℕ ⊆ ℂ) | |
5 | 1, 3, 4 | mp2an 690 | 1 ⊢ ℕ ⊆ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 (class class class)co 7155 ℂcc 10534 1c1 10537 + caddc 10539 ℕcn 11637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-1cn 10594 ax-addcl 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-nn 11638 |
This theorem is referenced by: nnex 11643 nncn 11645 nncnd 11653 nn0sscn 11901 nn0addcl 11931 nn0mulcl 11932 dfz2 11999 nnexpcl 13441 fprodnncl 15308 nnrisefaccl 15372 znnen 15564 wunndx 16503 cmetcaulem 23890 dvdsmulf1o 25770 fsumdvdsmul 25771 esumcvg 31345 eulerpartlemgs2 31638 fsum2dsub 31878 reprsuc 31886 nndivsub 33805 fsumnncl 41850 nnsgrpmgm 44082 nnsgrp 44083 nnsgrpnmnd 44084 |
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