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 10002 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | |
4 | 1, 2, 3 | syl2anc 691 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ℝcr 9814 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: add20 10419 eqord1 10435 msq11 10803 supadd 10868 supmul 10872 suprzcl 11333 uzwo3 11659 flid 12471 flval3 12478 gcd0id 15078 gcdneg 15081 bezoutlem4 15097 gcdzeq 15109 lcmneg 15154 coprmgcdb 15200 qredeq 15209 pcidlem 15414 pcgcd1 15419 4sqlem17 15503 0ram 15562 ram0 15564 mndodconglem 17783 sylow1lem5 17840 zntoslem 19724 cnmpt2pc 22535 ovolsca 23090 ismbl2 23102 voliunlem2 23126 dyadmaxlem 23171 mbfi1fseqlem4 23291 itg2cnlem1 23334 ditgneg 23427 rolle 23557 dvivthlem1 23575 plyeq0lem 23770 dgreq 23804 coemulhi 23814 dgradd2 23828 dgrmul 23830 plydiveu 23857 vieta1lem2 23870 pilem3 24011 zabsle1 24821 ostth2 25126 brbtwn2 25585 axcontlem8 25651 nmophmi 28274 leoptri 28379 2sqmod 28979 fzto1st1 29183 ballotlemfc0 29881 ballotlemfcc 29882 poimirlem23 32602 rmspecfund 36492 ubelsupr 38202 lefldiveq 38446 wallispilem3 38960 fourierdlem6 39006 fourierdlem42 39042 fourierdlem50 39049 fourierdlem52 39051 fourierdlem54 39053 fourierdlem79 39078 fourierdlem102 39101 fourierdlem114 39113 lighneallem2 40061 2ffzoeq 40361 |
Copyright terms: Public domain | W3C validator |