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Mirrors > Home > ILE Home > Th. List > nn0ssre | GIF version |
Description: Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
nn0ssre | ⊢ ℕ0 ⊆ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-n0 7958 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnssre 7699 | . . 3 ⊢ ℕ ⊆ ℝ | |
3 | 0re 6825 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | snssi 3499 | . . . 4 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ) | |
5 | 3, 4 | ax-mp 7 | . . 3 ⊢ {0} ⊆ ℝ |
6 | 2, 5 | unssi 3112 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℝ |
7 | 1, 6 | eqsstri 2969 | 1 ⊢ ℕ0 ⊆ ℝ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1390 ∪ cun 2909 ⊆ wss 2911 {csn 3367 ℝcr 6710 0cc0 6711 ℕcn 7695 ℕ0cn0 7957 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-cnex 6774 ax-resscn 6775 ax-1re 6777 ax-addrcl 6780 ax-rnegex 6792 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-int 3607 df-inn 7696 df-n0 7958 |
This theorem is referenced by: nn0sscn 7962 nn0re 7966 nn0rei 7968 nn0red 8012 |
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