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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 11301 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
2 | 1 | biancomd 463 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵))) |
3 | lenlt 11296 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | lenlt 11296 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) | |
5 | 4 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) |
6 | 3, 5 | anbi12d 630 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵))) |
7 | 2, 6 | bitr4d 282 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 ℝcr 11111 < clt 11252 ≤ cle 11253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 |
This theorem is referenced by: eqlelt 11305 eqlei 11328 eqlei2 11329 letri3i 11334 letri3d 11360 lesub0 11735 eqord1 11746 lbreu 12168 nnle1eq1 12246 nn0le0eq0 12504 zextle 12639 uz11 12851 uzin 12866 uzwo 12899 qsqueeze 13186 elfz1eq 13518 faclbnd4lem4 14261 swrdccat3blem 14695 repswswrd 14740 sqeqd 15119 max0add 15263 fsum00 15750 reef11 16069 dvdsabseq 16263 nn0seqcvgd 16514 infpnlem1 16852 gzrngunit 21327 psrbaglesupp 21818 psrbaglesuppOLD 21819 nmoeq0 24608 oprpiece1res2 24832 pcoval2 24898 minveclem7 25318 pjthlem1 25320 iblposlem 25676 dvferm 25875 dveq0 25888 dv11cn 25889 fta1blem 26060 dgrco 26165 aalioulem3 26224 logf1o2 26539 cxpsqrtlem 26591 ang180lem3 26698 chpeq0 27096 chteq0 27097 lgsdir 27220 lgsabs1 27224 minvecolem7 30645 pjhthlem1 31153 pjnormssi 31930 hstles 31993 stge1i 32000 stle0i 32001 stlesi 32003 cdj3lem1 32196 derangen 34691 bfplem2 37204 bfp 37205 acongeq 42305 jm2.26lem3 42323 dvconstbi 43674 zgeltp1eq 46594 zgtp1leeq 47482 |
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