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Mirrors > Home > ILE Home > Th. List > ressxr | GIF version |
Description: The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
ressxr | ⊢ ℝ ⊆ ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 3290 | . 2 ⊢ ℝ ⊆ (ℝ ∪ {+∞, -∞}) | |
2 | df-xr 7958 | . 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
3 | 1, 2 | sseqtrri 3182 | 1 ⊢ ℝ ⊆ ℝ* |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3119 ⊆ wss 3121 {cpr 3584 ℝcr 7773 +∞cpnf 7951 -∞cmnf 7952 ℝ*cxr 7953 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-xr 7958 |
This theorem is referenced by: rexpssxrxp 7964 rexr 7965 0xr 7966 rexrd 7969 ltrelxr 7980 iooval2 9872 fzval2 9968 seq3coll 10777 summodclem2a 11344 prodmodclem2a 11539 ismet2 13148 qtopbas 13316 tgqioo 13341 |
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