Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 0e0icopnf | GIF version |
Description: 0 is a member of (0[,)+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0e0icopnf | ⊢ 0 ∈ (0[,)+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7907 | . 2 ⊢ 0 ∈ ℝ | |
2 | 0le0 8954 | . 2 ⊢ 0 ≤ 0 | |
3 | elrege0 9920 | . 2 ⊢ (0 ∈ (0[,)+∞) ↔ (0 ∈ ℝ ∧ 0 ≤ 0)) | |
4 | 1, 2, 3 | mpbir2an 937 | 1 ⊢ 0 ∈ (0[,)+∞) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 class class class wbr 3987 (class class class)co 5850 ℝcr 7760 0cc0 7761 +∞cpnf 7938 ≤ cle 7942 [,)cico 9834 |
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-in1 609 ax-in2 610 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-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1re 7855 ax-addrcl 7858 ax-rnegex 7870 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-ico 9838 |
This theorem is referenced by: fsumge0 11409 |
Copyright terms: Public domain | W3C validator |