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Mirrors > Home > ILE Home > Th. List > ax0lt1 | GIF version |
Description: 0 is less than 1. Axiom
for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-0lt1 7352.
The version of this axiom in the Metamath Proof Explorer reads 1 ≠ 0; here we change it to 0 <ℝ 1. The proof of 0 <ℝ 1 from 1 ≠ 0 in the Metamath Proof Explorer (accessed 12-Jan-2020) relies on real number trichotomy. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax0lt1 | ⊢ 0 <ℝ 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1sr 7212 | . . 3 ⊢ 0R <R 1R | |
2 | ltresr 7277 | . . 3 ⊢ (〈0R, 0R〉 <ℝ 〈1R, 0R〉 ↔ 0R <R 1R) | |
3 | 1, 2 | mpbir 144 | . 2 ⊢ 〈0R, 0R〉 <ℝ 〈1R, 0R〉 |
4 | df-0 7258 | . 2 ⊢ 0 = 〈0R, 0R〉 | |
5 | df-1 7259 | . 2 ⊢ 1 = 〈1R, 0R〉 | |
6 | 3, 4, 5 | 3brtr4i 3839 | 1 ⊢ 0 <ℝ 1 |
Colors of variables: wff set class |
Syntax hints: 〈cop 3425 class class class wbr 3811 0Rc0r 6758 1Rc1r 6759 <R cltr 6763 0cc0 7251 1c1 7252 <ℝ cltrr 7255 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-eprel 4079 df-id 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-recs 6000 df-irdg 6065 df-1o 6111 df-2o 6112 df-oadd 6115 df-omul 6116 df-er 6220 df-ec 6222 df-qs 6226 df-ni 6764 df-pli 6765 df-mi 6766 df-lti 6767 df-plpq 6804 df-mpq 6805 df-enq 6807 df-nqqs 6808 df-plqqs 6809 df-mqqs 6810 df-1nqqs 6811 df-rq 6812 df-ltnqqs 6813 df-enq0 6884 df-nq0 6885 df-0nq0 6886 df-plq0 6887 df-mq0 6888 df-inp 6926 df-i1p 6927 df-iplp 6928 df-iltp 6930 df-enr 7173 df-nr 7174 df-ltr 7177 df-0r 7178 df-1r 7179 df-0 7258 df-1 7259 df-r 7261 df-lt 7264 |
This theorem is referenced by: (None) |
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