Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 1ne0sr | Structured version Visualization version GIF version |
Description: 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1ne0sr | ⊢ ¬ 1R = 0R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltsosr 10705 | . . 3 ⊢ <R Or R | |
2 | 1sr 10692 | . . 3 ⊢ 1R ∈ R | |
3 | sonr 5488 | . . 3 ⊢ (( <R Or R ∧ 1R ∈ R) → ¬ 1R <R 1R) | |
4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ ¬ 1R <R 1R |
5 | 0lt1sr 10706 | . . 3 ⊢ 0R <R 1R | |
6 | breq1 5053 | . . 3 ⊢ (1R = 0R → (1R <R 1R ↔ 0R <R 1R)) | |
7 | 5, 6 | mpbiri 261 | . 2 ⊢ (1R = 0R → 1R <R 1R) |
8 | 4, 7 | mto 200 | 1 ⊢ ¬ 1R = 0R |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2110 class class class wbr 5050 Or wor 5464 Rcnr 10476 0Rc0r 10477 1Rc1r 10478 <R cltr 10482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-inf2 9253 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-oadd 8203 df-omul 8204 df-er 8388 df-ec 8390 df-qs 8394 df-ni 10483 df-pli 10484 df-mi 10485 df-lti 10486 df-plpq 10519 df-mpq 10520 df-ltpq 10521 df-enq 10522 df-nq 10523 df-erq 10524 df-plq 10525 df-mq 10526 df-1nq 10527 df-rq 10528 df-ltnq 10529 df-np 10592 df-1p 10593 df-plp 10594 df-ltp 10596 df-enr 10666 df-nr 10667 df-ltr 10670 df-0r 10671 df-1r 10672 |
This theorem is referenced by: ax1ne0 10771 |
Copyright terms: Public domain | W3C validator |