Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7719.
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 7566 | . . 3 ⊢ 0R <R 1R | |
2 | ltresr 7640 | . . 3 ⊢ (〈0R, 0R〉 <ℝ 〈1R, 0R〉 ↔ 0R <R 1R) | |
3 | 1, 2 | mpbir 145 | . 2 ⊢ 〈0R, 0R〉 <ℝ 〈1R, 0R〉 |
4 | df-0 7620 | . 2 ⊢ 0 = 〈0R, 0R〉 | |
5 | df-1 7621 | . 2 ⊢ 1 = 〈1R, 0R〉 | |
6 | 3, 4, 5 | 3brtr4i 3953 | 1 ⊢ 0 <ℝ 1 |
Colors of variables: wff set class |
Syntax hints: 〈cop 3525 class class class wbr 3924 0Rc0r 7099 1Rc1r 7100 <R cltr 7104 0cc0 7613 1c1 7614 <ℝ cltrr 7617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-2o 6307 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-enq0 7225 df-nq0 7226 df-0nq0 7227 df-plq0 7228 df-mq0 7229 df-inp 7267 df-i1p 7268 df-iplp 7269 df-iltp 7271 df-enr 7527 df-nr 7528 df-ltr 7531 df-0r 7532 df-1r 7533 df-0 7620 df-1 7621 df-r 7623 df-lt 7626 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |