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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 9175 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnssre 8921 | . . 3 ⊢ ℕ ⊆ ℝ | |
3 | 0re 7956 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | snssi 3736 | . . . 4 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ {0} ⊆ ℝ |
6 | 2, 5 | unssi 3310 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℝ |
7 | 1, 6 | eqsstri 3187 | 1 ⊢ ℕ0 ⊆ ℝ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 ∪ cun 3127 ⊆ wss 3129 {csn 3592 ℝcr 7809 0cc0 7810 ℕcn 8917 ℕ0cn0 9174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4121 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 ax-rnegex 7919 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-int 3845 df-inn 8918 df-n0 9175 |
This theorem is referenced by: nn0sscn 9179 nn0re 9183 nn0rei 9185 nn0red 9228 |
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