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Mirrors > Home > MPE Home > Th. List > leltne | Structured version Visualization version GIF version |
Description: 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by NM, 27-Jul-1999.) |
Ref | Expression |
---|---|
leltne | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 10000 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
2 | simpl 472 | . . . . . . 7 ⊢ ((¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴) → ¬ 𝐴 < 𝐵) | |
3 | 1, 2 | syl6bi 242 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵)) |
4 | 3 | adantr 480 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵)) |
5 | leloe 10003 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
6 | 5 | biimpa 500 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) |
7 | 6 | ord 391 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (¬ 𝐴 < 𝐵 → 𝐴 = 𝐵)) |
8 | 4, 7 | impbid 201 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐴 = 𝐵 ↔ ¬ 𝐴 < 𝐵)) |
9 | 8 | necon2abid 2824 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐴 ≠ 𝐵)) |
10 | necom 2835 | . . 3 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵) | |
11 | 9, 10 | syl6bbr 277 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) |
12 | 11 | 3impa 1251 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ℝcr 9814 < clt 9953 ≤ 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: leltned 10069 nngt1ne1 10924 nn01to3 11657 gcdn0gt0 15077 isprm3 15234 iundisj2 23124 clwlkisclwwlklem2a4 26312 norm-i 27370 cnlnadjlem7 28316 iundisj2f 28785 iundisj2fi 28943 fmul01lt1lem1 38651 icccncfext 38773 iblcncfioo 38870 clwlkclwwlklem2a4 41206 |
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