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Theorem xrltlen 12226

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 12219	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
2		ioran 1007	. . . 4 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴))
3		ancom 453	. . . 4 ⊢ ((¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵))
4	2, 3	bitri 267	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵))
5	1, 4	syl6bb 279	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵)))
6		xrlenlt 10393	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
7		nesym 3027	. . . 4 ⊢ (𝐵 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐵)
8	7	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐵))
9	6, 8	anbi12d 625	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵)))
10	5, 9	bitr4d 274	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 ∨ wo 874 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 class class class wbr 4843 ℝ^*cxr 10362 < clt 10363 ≤ cle 10364

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-pre-lttri 10298 ax-pre-lttrn 10299

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369

This theorem is referenced by: dflt2 12228 hashgt0 13427

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