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Mirrors > Home > MPE Home > Th. List > letri3d | Structured version Visualization version GIF version |
Description: Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
letri3d | ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | letri3 10325 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | |
4 | 1, 2, 3 | syl2anc 573 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ℝcr 10137 ≤ cle 10277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-pre-lttri 10212 ax-pre-lttrn 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 |
This theorem is referenced by: add20 10742 eqord1 10758 msq11 11126 supadd 11193 supmul 11197 suprzcl 11659 uzwo3 11986 flid 12817 flval3 12824 gcd0id 15448 gcdneg 15451 bezoutlem4 15467 gcdzeq 15479 lcmneg 15524 coprmgcdb 15570 qredeq 15578 pcidlem 15783 pcgcd1 15788 4sqlem17 15872 0ram 15931 ram0 15933 mndodconglem 18167 sylow1lem5 18224 zntoslem 20120 cnmpt2pc 22947 ovolsca 23503 ismbl2 23515 voliunlem2 23539 dyadmaxlem 23585 mbfi1fseqlem4 23705 itg2cnlem1 23748 ditgneg 23841 rolle 23973 dvivthlem1 23991 plyeq0lem 24186 dgreq 24220 coemulhi 24230 dgradd2 24244 dgrmul 24246 plydiveu 24273 vieta1lem2 24286 pilem3 24427 pilem3OLD 24428 zabsle1 25242 ostth2 25547 brbtwn2 26006 axcontlem8 26072 nmophmi 29230 leoptri 29335 2sqmod 29988 fzto1st1 30192 ballotlemfc0 30894 ballotlemfcc 30895 poimirlem23 33765 rmspecfund 38000 ubelsupr 39701 lefldiveq 40023 wallispilem3 40801 fourierdlem6 40847 fourierdlem42 40883 fourierdlem50 40890 fourierdlem52 40892 fourierdlem54 40894 fourierdlem79 40919 fourierdlem102 40942 fourierdlem114 40954 2ffzoeq 41866 lighneallem2 42051 |
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