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Theorem xrleltne 13059

Description: 'Less than or equal to' implies 'less than' is not 'equals', for extended reals. (Contributed by NM, 9-Feb-2006.)

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

**Proof of Theorem xrleltne**
Step	Hyp	Ref	Expression
1		xrlttri3 13057	. . . . . . 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		xrleloe 13058	. . . . . . 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 2974	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐴 ≠ 𝐵))
10		necom 2985	. . 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 1541 ∈ wcel 2113 ≠ wne 2932 class class class wbr 5098 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-pre-lttri 11100 ax-pre-lttrn 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172

This theorem is referenced by: xrsdsreclblem 21367 nmopgt0 31987 elicc3 36511 xrleneltd 45564 icoiccdif 45766

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