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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 10997 | . . . . . 6 ⊢ 1P ∈ P | |
2 | addclpr 11000 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
3 | 1, 1, 2 | mp2an 691 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
4 | ltaddpr 11016 | . . . . 5 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → (1P +P 1P)<P ((1P +P 1P) +P 1P)) | |
5 | 3, 1, 4 | mp2an 691 | . . . 4 ⊢ (1P +P 1P)<P ((1P +P 1P) +P 1P) |
6 | addcompr 11003 | . . . 4 ⊢ (1P +P (1P +P 1P)) = ((1P +P 1P) +P 1P) | |
7 | 5, 6 | breqtrri 5171 | . . 3 ⊢ (1P +P 1P)<P (1P +P (1P +P 1P)) |
8 | ltsrpr 11059 | . . 3 ⊢ ([〈1P, 1P〉] ~R <R [〈(1P +P 1P), 1P〉] ~R ↔ (1P +P 1P)<P (1P +P (1P +P 1P))) | |
9 | 7, 8 | mpbir 230 | . 2 ⊢ [〈1P, 1P〉] ~R <R [〈(1P +P 1P), 1P〉] ~R |
10 | df-0r 11042 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
11 | df-1r 11043 | . 2 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
12 | 9, 10, 11 | 3brtr4i 5174 | 1 ⊢ 0R <R 1R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 〈cop 4630 class class class wbr 5144 (class class class)co 7396 [cec 8689 Pcnp 10841 1Pc1p 10842 +P cpp 10843 <P cltp 10845 ~R cer 10846 0Rc0r 10848 1Rc1r 10849 <R cltr 10853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-inf2 9623 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-oadd 8457 df-omul 8458 df-er 8691 df-ec 8693 df-qs 8697 df-ni 10854 df-pli 10855 df-mi 10856 df-lti 10857 df-plpq 10890 df-mpq 10891 df-ltpq 10892 df-enq 10893 df-nq 10894 df-erq 10895 df-plq 10896 df-mq 10897 df-1nq 10898 df-rq 10899 df-ltnq 10900 df-np 10963 df-1p 10964 df-plp 10965 df-ltp 10967 df-enr 11037 df-nr 11038 df-ltr 11041 df-0r 11042 df-1r 11043 |
This theorem is referenced by: 1ne0sr 11078 supsrlem 11093 |
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