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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 12268 and ax-resscn 11210 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 11211 | . 2 ⊢ 1 ∈ ℂ | |
2 | peano2cn 11431 | . . 3 ⊢ (𝑥 ∈ ℂ → (𝑥 + 1) ∈ ℂ) | |
3 | 2 | rgen 3061 | . 2 ⊢ ∀𝑥 ∈ ℂ (𝑥 + 1) ∈ ℂ |
4 | peano5nni 12267 | . 2 ⊢ ((1 ∈ ℂ ∧ ∀𝑥 ∈ ℂ (𝑥 + 1) ∈ ℂ) → ℕ ⊆ ℂ) | |
5 | 1, 3, 4 | mp2an 692 | 1 ⊢ ℕ ⊆ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 (class class class)co 7431 ℂcc 11151 1c1 11154 + caddc 11156 ℕcn 12264 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 |
This theorem is referenced by: nnex 12270 nncn 12272 nncnd 12280 nn0sscn 12529 nn0addcl 12559 nn0mulcl 12560 dfz2 12630 nnexpcl 14112 fprodnncl 15988 nnrisefaccl 16052 znnen 16245 wunndx 17229 cmetcaulem 25336 mpodvdsmulf1o 27252 fsumdvdsmul 27253 dvdsmulf1o 27254 fsumdvdsmulOLD 27255 esumcvg 34067 eulerpartlemgs2 34362 fsum2dsub 34601 reprsuc 34609 nndivsub 36440 fsumnncl 45528 nnsgrpmgm 48020 nnsgrp 48021 nnsgrpnmnd 48022 |
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