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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 10431 | . . . . . 6 ⊢ 1P ∈ P | |
2 | addclpr 10434 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
3 | 1, 1, 2 | mp2an 690 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
4 | ltaddpr 10450 | . . . . 5 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → (1P +P 1P)<P ((1P +P 1P) +P 1P)) | |
5 | 3, 1, 4 | mp2an 690 | . . . 4 ⊢ (1P +P 1P)<P ((1P +P 1P) +P 1P) |
6 | addcompr 10437 | . . . 4 ⊢ (1P +P (1P +P 1P)) = ((1P +P 1P) +P 1P) | |
7 | 5, 6 | breqtrri 5086 | . . 3 ⊢ (1P +P 1P)<P (1P +P (1P +P 1P)) |
8 | ltsrpr 10493 | . . 3 ⊢ ([〈1P, 1P〉] ~R <R [〈(1P +P 1P), 1P〉] ~R ↔ (1P +P 1P)<P (1P +P (1P +P 1P))) | |
9 | 7, 8 | mpbir 233 | . 2 ⊢ [〈1P, 1P〉] ~R <R [〈(1P +P 1P), 1P〉] ~R |
10 | df-0r 10476 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
11 | df-1r 10477 | . 2 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
12 | 9, 10, 11 | 3brtr4i 5089 | 1 ⊢ 0R <R 1R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 〈cop 4567 class class class wbr 5059 (class class class)co 7150 [cec 8281 Pcnp 10275 1Pc1p 10276 +P cpp 10277 <P cltp 10279 ~R cer 10280 0Rc0r 10282 1Rc1r 10283 <R cltr 10287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-rq 10333 df-ltnq 10334 df-np 10397 df-1p 10398 df-plp 10399 df-ltp 10401 df-enr 10471 df-nr 10472 df-ltr 10475 df-0r 10476 df-1r 10477 |
This theorem is referenced by: 1ne0sr 10512 supsrlem 10527 |
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