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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 12254 and ax-resscn 11202 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 11203 | . 2 ⊢ 1 ∈ ℂ | |
2 | peano2cn 11423 | . . 3 ⊢ (𝑥 ∈ ℂ → (𝑥 + 1) ∈ ℂ) | |
3 | 2 | rgen 3052 | . 2 ⊢ ∀𝑥 ∈ ℂ (𝑥 + 1) ∈ ℂ |
4 | peano5nni 12253 | . 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 2098 ∀wral 3050 ⊆ wss 3944 (class class class)co 7419 ℂcc 11143 1c1 11146 + caddc 11148 ℕcn 12250 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 ax-1cn 11203 ax-addcl 11205 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-nn 12251 |
This theorem is referenced by: nnex 12256 nncn 12258 nncnd 12266 nn0sscn 12515 nn0addcl 12545 nn0mulcl 12546 dfz2 12615 nnexpcl 14080 fprodnncl 15940 nnrisefaccl 16004 znnen 16197 wunndx 17172 cmetcaulem 25265 mpodvdsmulf1o 27176 fsumdvdsmul 27177 dvdsmulf1o 27178 fsumdvdsmulOLD 27179 esumcvg 33838 eulerpartlemgs2 34133 fsum2dsub 34372 reprsuc 34380 nndivsub 36074 fsumnncl 45100 nnsgrpmgm 47426 nnsgrp 47427 nnsgrpnmnd 47428 |
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