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Theorem xrltlen 13152

Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)

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

**Proof of Theorem xrltlen**
Step	Hyp	Ref	Expression
1		xrlttri 13145	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
2		ioran 982	. . . 4 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴))
3	2	biancomi 462	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵))
4	1, 3	bitrdi 287	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵)))
5		xrlenlt 11304	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
6		nesym 2993	. . . 4 ⊢ (𝐵 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐵)
7	6	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐵))
8	5, 7	anbi12d 631	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵)))
9	4, 8	bitr4d 282	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 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 class class class wbr 5143 ℝ^*cxr 11272 < clt 11273 ≤ cle 11274

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-pre-lttri 11207 ax-pre-lttrn 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279

This theorem is referenced by: dflt2 13154 hashgt0 14374

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