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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 9837 | . . . . . 6 ⊢ 1P ∈ P | |
2 | addclpr 9840 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
3 | 1, 1, 2 | mp2an 708 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
4 | ltaddpr 9856 | . . . . 5 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → (1P +P 1P)<P ((1P +P 1P) +P 1P)) | |
5 | 3, 1, 4 | mp2an 708 | . . . 4 ⊢ (1P +P 1P)<P ((1P +P 1P) +P 1P) |
6 | addcompr 9843 | . . . 4 ⊢ (1P +P (1P +P 1P)) = ((1P +P 1P) +P 1P) | |
7 | 5, 6 | breqtrri 4680 | . . 3 ⊢ (1P +P 1P)<P (1P +P (1P +P 1P)) |
8 | ltsrpr 9898 | . . 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 9882 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
11 | df-1r 9883 | . 2 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
12 | 9, 10, 11 | 3brtr4i 4683 | 1 ⊢ 0R <R 1R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 〈cop 4183 class class class wbr 4653 (class class class)co 6650 [cec 7740 Pcnp 9681 1Pc1p 9682 +P cpp 9683 <P cltp 9685 ~R cer 9686 0Rc0r 9688 1Rc1r 9689 <R cltr 9693 |
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-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-ni 9694 df-pli 9695 df-mi 9696 df-lti 9697 df-plpq 9730 df-mpq 9731 df-ltpq 9732 df-enq 9733 df-nq 9734 df-erq 9735 df-plq 9736 df-mq 9737 df-1nq 9738 df-rq 9739 df-ltnq 9740 df-np 9803 df-1p 9804 df-plp 9805 df-ltp 9807 df-enr 9877 df-nr 9878 df-ltr 9881 df-0r 9882 df-1r 9883 |
This theorem is referenced by: 1ne0sr 9917 supsrlem 9932 |
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