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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 10724 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
2 | 1 | biancomd 466 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵))) |
3 | lenlt 10719 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | lenlt 10719 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) | |
5 | 4 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) |
6 | 3, 5 | anbi12d 632 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵))) |
7 | 2, 6 | bitr4d 284 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ℝcr 10536 < clt 10675 ≤ cle 10676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 |
This theorem is referenced by: eqlelt 10728 eqlei 10750 eqlei2 10751 letri3i 10756 letri3d 10782 lesub0 11157 eqord1 11168 lbreu 11591 nnle1eq1 11668 nn0le0eq0 11926 zextle 12056 uz11 12268 uzin 12279 uzwo 12312 qsqueeze 12595 elfz1eq 12919 faclbnd4lem4 13657 swrdccat3blem 14101 repswswrd 14146 sqeqd 14525 max0add 14670 fsum00 15153 reef11 15472 dvdsabseq 15663 nn0seqcvgd 15914 infpnlem1 16246 psrbaglesupp 20148 gzrngunit 20611 nmoeq0 23345 oprpiece1res2 23556 pcoval2 23620 minveclem7 24038 pjthlem1 24040 iblposlem 24392 dvferm 24585 dveq0 24597 dv11cn 24598 fta1blem 24762 dgrco 24865 aalioulem3 24923 logf1o2 25233 cxpsqrtlem 25285 ang180lem3 25389 chpeq0 25784 chteq0 25785 lgsdir 25908 lgsabs1 25912 minvecolem7 28660 pjhthlem1 29168 pjnormssi 29945 hstles 30008 stge1i 30015 stle0i 30016 stlesi 30018 cdj3lem1 30211 derangen 32419 bfplem2 35116 bfp 35117 acongeq 39600 jm2.26lem3 39618 dvconstbi 40686 zgeltp1eq 43529 zgtp1leeq 44596 |
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