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Mirrors > Home > MPE Home > Th. List > nnssre | Structured version Visualization version GIF version |
Description: The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) |
Ref | Expression |
---|---|
nnssre | ⊢ ℕ ⊆ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 11252 | . 2 ⊢ 1 ∈ ℝ | |
2 | peano2re 11425 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ) | |
3 | 2 | rgen 3053 | . 2 ⊢ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ |
4 | peano5nni 12258 | . 2 ⊢ ((1 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ) → ℕ ⊆ ℝ) | |
5 | 1, 3, 4 | mp2an 690 | 1 ⊢ ℕ ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ∀wral 3051 ⊆ wss 3946 (class class class)co 7413 ℝcr 11145 1c1 11147 + caddc 11149 ℕcn 12255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7735 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-i2m1 11214 ax-1ne0 11215 ax-rrecex 11218 ax-cnre 11219 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-om 7866 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-nn 12256 |
This theorem is referenced by: nnre 12262 dfnn3 12269 nnred 12270 nnunb 12511 nn0ssre 12519 isercolllem1 15661 isercolllem2 15662 isercoll 15664 o1fsum 15809 ruc 16237 prmgaplem3 17047 prmgaplem4 17048 gsumval3 19898 ovolctb2 25506 ovolicc2lem3 25533 ovolicc2lem4 25534 iundisj2 25563 iundisj2f 32507 ssnnssfz 32689 iundisjfi 32698 iundisj2fi 32699 xrsmulgzz 32889 ballotlemsup 34348 reprlt 34475 reprgt 34477 erdszelem5 35033 erdszelem7 35035 erdszelem8 35036 incsequz2 37460 aks6d1c2 41839 sticksstones1 41855 stoweidlem34 45688 fourierdlem31 45792 prmdvdsfmtnof1lem1 47189 prmdvdsfmtnof 47191 |
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