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Mirrors > Home > ILE Home > Th. List > 0lepnf | GIF version |
Description: 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0lepnf | ⊢ 0 ≤ +∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7966 | . 2 ⊢ 0 ∈ ℝ* | |
2 | pnfge 9746 | . 2 ⊢ (0 ∈ ℝ* → 0 ≤ +∞) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 0 ≤ +∞ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 class class class wbr 3989 0cc0 7774 +∞cpnf 7951 ℝ*cxr 7953 ≤ cle 7955 |
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 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871 ax-rnegex 7883 |
This theorem depends on definitions: df-bi 116 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-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 |
This theorem is referenced by: nn0pnfge0 9748 xsubge0 9838 pcge0 12266 |
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