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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 8936 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnssre 8688 | . . 3 ⊢ ℕ ⊆ ℝ | |
3 | 0re 7734 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | snssi 3634 | . . . 4 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ {0} ⊆ ℝ |
6 | 2, 5 | unssi 3221 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℝ |
7 | 1, 6 | eqsstri 3099 | 1 ⊢ ℕ0 ⊆ ℝ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 ∪ cun 3039 ⊆ wss 3041 {csn 3497 ℝcr 7587 0cc0 7588 ℕcn 8684 ℕ0cn0 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 ax-rnegex 7697 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-int 3742 df-inn 8685 df-n0 8936 |
This theorem is referenced by: nn0sscn 8940 nn0re 8944 nn0rei 8946 nn0red 8989 |
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