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Mirrors > Home > ILE Home > Th. List > 1ne0sr | GIF version |
Description: 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) |
Ref | Expression |
---|---|
1ne0sr | ⊢ ¬ 1R = 0R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltsosr 7726 | . . 3 ⊢ <R Or R | |
2 | 1sr 7713 | . . 3 ⊢ 1R ∈ R | |
3 | sonr 4302 | . . 3 ⊢ (( <R Or R ∧ 1R ∈ R) → ¬ 1R <R 1R) | |
4 | 1, 2, 3 | mp2an 424 | . 2 ⊢ ¬ 1R <R 1R |
5 | 0lt1sr 7727 | . . 3 ⊢ 0R <R 1R | |
6 | breq1 3992 | . . 3 ⊢ (1R = 0R → (1R <R 1R ↔ 0R <R 1R)) | |
7 | 5, 6 | mpbiri 167 | . 2 ⊢ (1R = 0R → 1R <R 1R) |
8 | 4, 7 | mto 657 | 1 ⊢ ¬ 1R = 0R |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1348 ∈ wcel 2141 class class class wbr 3989 Or wor 4280 Rcnr 7259 0Rc0r 7260 1Rc1r 7261 <R cltr 7265 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-2o 6396 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-enq0 7386 df-nq0 7387 df-0nq0 7388 df-plq0 7389 df-mq0 7390 df-inp 7428 df-i1p 7429 df-iplp 7430 df-iltp 7432 df-enr 7688 df-nr 7689 df-ltr 7692 df-0r 7693 df-1r 7694 |
This theorem is referenced by: (None) |
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