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Mirrors > Home > MPE Home > Th. List > letri3 | Structured version Visualization version GIF version |
Description: Trichotomy law. (Contributed by NM, 14-May-1999.) |
Ref | Expression |
---|---|
letri3 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 10121 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
2 | ancom 466 | . . 3 ⊢ ((¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵) ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | |
3 | 1, 2 | syl6bbr 278 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵))) |
4 | lenlt 10116 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
5 | lenlt 10116 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) | |
6 | 5 | ancoms 469 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) |
7 | 4, 6 | anbi12d 747 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵))) |
8 | 3, 7 | bitr4d 271 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ℝcr 9935 < clt 10074 ≤ cle 10075 |
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-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
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-nel 2898 df-ral 2917 df-rex 2918 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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 |
This theorem is referenced by: eqlelt 10125 eqlei 10147 eqlei2 10148 letri3i 10153 letri3d 10179 lesub0 10545 eqord1 10556 lbreu 10973 nnle1eq1 11048 nn0le0eq0 11321 zextle 11450 uz11 11710 uzin 11720 uzwo 11751 qsqueeze 12032 elfz1eq 12352 faclbnd4lem4 13083 swrdccat3blem 13495 repswswrd 13531 sqeqd 13906 max0add 14050 fsum00 14530 reef11 14849 dvdsabseq 15035 nn0seqcvgd 15283 infpnlem1 15614 psrbaglesupp 19368 gzrngunit 19812 nmoeq0 22540 oprpiece1res2 22751 pcoval2 22816 minveclem7 23206 pjthlem1 23208 iblposlem 23558 dvferm 23751 dveq0 23763 dv11cn 23764 fta1blem 23928 dgrco 24031 aalioulem3 24089 logf1o2 24396 cxpsqrtlem 24448 ang180lem3 24541 chpeq0 24933 chteq0 24934 lgsdir 25057 lgsabs1 25061 minvecolem7 27739 pjhthlem1 28250 pjnormssi 29027 hstles 29090 stge1i 29097 stle0i 29098 stlesi 29100 cdj3lem1 29293 derangen 31154 bfplem2 33622 bfp 33623 acongeq 37550 jm2.26lem3 37568 dvconstbi 38533 zgeltp1eq 41318 zgtp1leeq 42311 |
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