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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltnltne | Structured version Visualization version GIF version |
Description: Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
Ref | Expression |
---|---|
ltnltne | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐵 = 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnle 10512 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | |
2 | leloe 10519 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ (𝐵 < 𝐴 ∨ 𝐵 = 𝐴))) | |
3 | 2 | ancoms 451 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ (𝐵 < 𝐴 ∨ 𝐵 = 𝐴))) |
4 | 3 | notbid 310 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 ≤ 𝐴 ↔ ¬ (𝐵 < 𝐴 ∨ 𝐵 = 𝐴))) |
5 | ioran 966 | . . 3 ⊢ (¬ (𝐵 < 𝐴 ∨ 𝐵 = 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐵 = 𝐴)) | |
6 | 5 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐵 < 𝐴 ∨ 𝐵 = 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐵 = 𝐴))) |
7 | 1, 4, 6 | 3bitrd 297 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐵 = 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2048 class class class wbr 4923 ℝcr 10326 < clt 10466 ≤ cle 10467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-resscn 10384 ax-pre-lttri 10401 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 |
This theorem is referenced by: (None) |
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