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Theorem leltne 11270

Description: 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by NM, 27-Jul-1999.)

**Assertion**
Ref	Expression
leltne	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴))

**Proof of Theorem leltne**
Step	Hyp	Ref	Expression
1		lttri3 11264	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
2		simpl 482	. . . . . . 7 ⊢ ((¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴) → ¬ 𝐴 < 𝐵)
3	1, 2	biimtrdi 253	. . . . . 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 11267	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
6	5	biimpa 476	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))
7	6	ord 864	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (¬ 𝐴 < 𝐵 → 𝐴 = 𝐵))
8	4, 7	impbid 212	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐴 = 𝐵 ↔ ¬ 𝐴 < 𝐵))
9	8	necon2abid 2968	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐴 ≠ 𝐵))
10		necom 2979	. . 3 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵)
11	9, 10	bitr4di 289	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴))
12	11	3impa 1109	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 class class class wbr 5110 ℝcr 11074 < clt 11215 ≤ cle 11216

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-resscn 11132 ax-pre-lttri 11149 ax-pre-lttrn 11150

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221

This theorem is referenced by: leltned 11334 nngt1ne1 12222 nn01to3 12907 gcdn0gt0 16495 isprm3 16660 iundisj2 25457 clwlkclwwlklem2a4 29933 norm-i 31065 cnlnadjlem7 32009 iundisj2f 32526 iundisj2fi 32727 fmul01lt1lem1 45589 icccncfext 45892 iblcncfioo 45983

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