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Mirrors > Home > MPE Home > Th. List > 0lt1sr | Structured version Visualization version GIF version |
Description: 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0lt1sr | ⊢ 0R <R 1R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pr 10029 | . . . . . 6 ⊢ 1P ∈ P | |
2 | addclpr 10032 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
3 | 1, 1, 2 | mp2an 710 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
4 | ltaddpr 10048 | . . . . 5 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → (1P +P 1P)<P ((1P +P 1P) +P 1P)) | |
5 | 3, 1, 4 | mp2an 710 | . . . 4 ⊢ (1P +P 1P)<P ((1P +P 1P) +P 1P) |
6 | addcompr 10035 | . . . 4 ⊢ (1P +P (1P +P 1P)) = ((1P +P 1P) +P 1P) | |
7 | 5, 6 | breqtrri 4831 | . . 3 ⊢ (1P +P 1P)<P (1P +P (1P +P 1P)) |
8 | ltsrpr 10090 | . . 3 ⊢ ([〈1P, 1P〉] ~R <R [〈(1P +P 1P), 1P〉] ~R ↔ (1P +P 1P)<P (1P +P (1P +P 1P))) | |
9 | 7, 8 | mpbir 221 | . 2 ⊢ [〈1P, 1P〉] ~R <R [〈(1P +P 1P), 1P〉] ~R |
10 | df-0r 10074 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
11 | df-1r 10075 | . 2 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
12 | 9, 10, 11 | 3brtr4i 4834 | 1 ⊢ 0R <R 1R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2139 〈cop 4327 class class class wbr 4804 (class class class)co 6813 [cec 7909 Pcnp 9873 1Pc1p 9874 +P cpp 9875 <P cltp 9877 ~R cer 9878 0Rc0r 9880 1Rc1r 9881 <R cltr 9885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-omul 7734 df-er 7911 df-ec 7913 df-qs 7917 df-ni 9886 df-pli 9887 df-mi 9888 df-lti 9889 df-plpq 9922 df-mpq 9923 df-ltpq 9924 df-enq 9925 df-nq 9926 df-erq 9927 df-plq 9928 df-mq 9929 df-1nq 9930 df-rq 9931 df-ltnq 9932 df-np 9995 df-1p 9996 df-plp 9997 df-ltp 9999 df-enr 10069 df-nr 10070 df-ltr 10073 df-0r 10074 df-1r 10075 |
This theorem is referenced by: 1ne0sr 10109 supsrlem 10124 |
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