| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 11346 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: add20 11775 eqord1 11791 msq11 12169 supadd 12236 supmul 12240 suprzcl 12698 uzwo3 12985 flid 13848 flval3 13855 gcd0id 16556 gcdneg 16559 bezoutlem4 16579 gcdzeq 16589 lcmneg 16640 coprmgcdb 16686 qredeq 16694 pcidlem 16910 pcgcd1 16915 4sqlem17 16999 0ram 17058 ram0 17060 mndodconglem 19559 sylow1lem5 19620 zntoslem 21575 cnmpopc 24955 ovolsca 25550 ismbl2 25562 voliunlem2 25586 dyadmaxlem 25632 mbfi1fseqlem4 25753 itg2cnlem1 25796 ditgneg 25892 rolle 26028 dvivthlem1 26047 plyeq0lem 26249 dgreq 26283 coemulhi 26293 dgradd2 26308 dgrmul 26310 plydiveu 26340 vieta1lem2 26353 pilem3 26497 recxpf1lem 26771 zabsle1 27340 2sqmod 27480 ostth2 27681 brbtwn2 28920 axcontlem8 28986 nmophmi 32050 leoptri 32155 fzto1st1 33122 ballotlemfc0 34495 ballotlemfcc 34496 0nn0m1nnn0 35118 poimirlem23 37650 unitscyglem1 42196 rmspecfund 42920 ubelsupr 45025 lefldiveq 45304 wallispilem3 46082 fourierdlem6 46128 fourierdlem42 46164 fourierdlem50 46171 fourierdlem52 46173 fourierdlem54 46175 fourierdlem79 46200 fourierdlem102 46223 fourierdlem114 46235 2ffzoeq 47339 lighneallem2 47593 |
| Copyright terms: Public domain | W3C validator |