![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11259 | . 2 ⊢ 1 ∈ ℝ | |
2 | peano2re 11432 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ) | |
3 | 2 | rgen 3061 | . 2 ⊢ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ |
4 | peano5nni 12267 | . 2 ⊢ ((1 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (𝑥 + 1) ∈ ℝ) → ℕ ⊆ ℝ) | |
5 | 1, 3, 4 | mp2an 692 | 1 ⊢ ℕ ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 (class class class)co 7431 ℝcr 11152 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-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-i2m1 11221 ax-1ne0 11222 ax-rrecex 11225 ax-cnre 11226 |
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: nnre 12271 dfnn3 12278 nnred 12279 nnunb 12520 nn0ssre 12528 isercolllem1 15698 isercolllem2 15699 isercoll 15701 o1fsum 15846 ruc 16276 prmgaplem3 17087 prmgaplem4 17088 gsumval3 19940 ovolctb2 25541 ovolicc2lem3 25568 ovolicc2lem4 25569 iundisj2 25598 iundisj2f 32610 ssnnssfz 32796 iundisjfi 32804 iundisj2fi 32805 xrsmulgzz 32994 ballotlemsup 34486 reprlt 34613 reprgt 34615 erdszelem5 35180 erdszelem7 35182 erdszelem8 35183 incsequz2 37736 aks6d1c2 42112 sticksstones1 42128 stoweidlem34 45990 fourierdlem31 46094 prmdvdsfmtnof1lem1 47509 prmdvdsfmtnof 47511 |
Copyright terms: Public domain | W3C validator |