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Mirrors > Home > MPE Home > Th. List > 0ltpnf | Structured version Visualization version GIF version |
Description: Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0ltpnf | ⊢ 0 < +∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10040 | . 2 ⊢ 0 ∈ ℝ | |
2 | ltpnf 11954 | . 2 ⊢ (0 ∈ ℝ → 0 < +∞) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 0 < +∞ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 class class class wbr 4653 ℝcr 9935 0cc0 9936 +∞cpnf 10071 < clt 10074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-iota 5851 df-fv 5896 df-ov 6653 df-pnf 10076 df-xr 10078 df-ltxr 10079 |
This theorem is referenced by: xmulgt0 12113 reltxrnmnf 12172 hashneq0 13155 hashge2el2dif 13262 sgnpnf 13833 pnfnei 21024 0bdop 28852 xlt2addrd 29523 xrge0mulc1cn 29987 pnfneige0 29997 lmxrge0 29998 mbfposadd 33457 ftc1anclem5 33489 fourierdlem111 40434 fouriersw 40448 |
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