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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 10123 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | |
4 | 1, 2, 3 | syl2anc 693 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ℝcr 9935 ≤ 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: add20 10540 eqord1 10556 msq11 10924 supadd 10991 supmul 10995 suprzcl 11457 uzwo3 11783 flid 12609 flval3 12616 gcd0id 15240 gcdneg 15243 bezoutlem4 15259 gcdzeq 15271 lcmneg 15316 coprmgcdb 15362 qredeq 15371 pcidlem 15576 pcgcd1 15581 4sqlem17 15665 0ram 15724 ram0 15726 mndodconglem 17960 sylow1lem5 18017 zntoslem 19905 cnmpt2pc 22727 ovolsca 23283 ismbl2 23295 voliunlem2 23319 dyadmaxlem 23365 mbfi1fseqlem4 23485 itg2cnlem1 23528 ditgneg 23621 rolle 23753 dvivthlem1 23771 plyeq0lem 23966 dgreq 24000 coemulhi 24010 dgradd2 24024 dgrmul 24026 plydiveu 24053 vieta1lem2 24066 pilem3 24207 zabsle1 25021 ostth2 25326 brbtwn2 25785 axcontlem8 25851 nmophmi 28890 leoptri 28995 2sqmod 29648 fzto1st1 29852 ballotlemfc0 30554 ballotlemfcc 30555 poimirlem23 33432 rmspecfund 37474 ubelsupr 39179 lefldiveq 39505 wallispilem3 40284 fourierdlem6 40330 fourierdlem42 40366 fourierdlem50 40373 fourierdlem52 40375 fourierdlem54 40377 fourierdlem79 40402 fourierdlem102 40425 fourierdlem114 40437 2ffzoeq 41338 lighneallem2 41523 |
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