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Mirrors > Home > MPE Home > Th. List > ltlen | Structured version Visualization version GIF version |
Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.) |
Ref | Expression |
---|---|
ltlen | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltle 11306 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | |
2 | ltne 11315 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | |
3 | 2 | ex 413 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐵 → 𝐵 ≠ 𝐴)) |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐵 ≠ 𝐴)) |
5 | 1, 4 | jcad 513 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴))) |
6 | leloe 11304 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
7 | df-ne 2941 | . . . . . 6 ⊢ (𝐵 ≠ 𝐴 ↔ ¬ 𝐵 = 𝐴) | |
8 | pm2.24 124 | . . . . . . 7 ⊢ (𝐵 = 𝐴 → (¬ 𝐵 = 𝐴 → 𝐴 < 𝐵)) | |
9 | 8 | eqcoms 2740 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (¬ 𝐵 = 𝐴 → 𝐴 < 𝐵)) |
10 | 7, 9 | biimtrid 241 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐵 ≠ 𝐴 → 𝐴 < 𝐵)) |
11 | 10 | jao1i 856 | . . . 4 ⊢ ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 ≠ 𝐴 → 𝐴 < 𝐵)) |
12 | 6, 11 | syl6bi 252 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → (𝐵 ≠ 𝐴 → 𝐴 < 𝐵))) |
13 | 12 | impd 411 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴) → 𝐴 < 𝐵)) |
14 | 5, 13 | impbid 211 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5148 ℝcr 11111 < clt 11252 ≤ cle 11253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 |
This theorem is referenced by: ltleni 11336 ltlend 11363 nn0lt2 12629 rpneg 13010 fzofzim 13683 elfznelfzob 13742 hashsdom 14345 cnpart 15191 oddprmgt2 16640 chfacfisf 22576 chfacfisfcpmat 22577 ang180lem2 26539 mumullem2 26908 lgsneg 27048 lgsdilem 27051 lgsdirprm 27058 2sqreultlem 27174 2sqreunnltlem 27177 axlowdimlem16 28470 unitdivcld 33167 zltp1ne 34385 poimirlem24 36815 itg2addnclem 36842 fzopredsuc 46330 iccpartiltu 46389 icceuelpartlem 46402 difmodm1lt 47296 |
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