xrlenlt - Intuitionistic Logic Explorer

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Theorem xrlenlt 8354

Description: 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)

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

**Proof of Theorem xrlenlt**
Step	Hyp	Ref	Expression
1		df-br 4115	. . 3 ⊢ (𝐴 ≤ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
2		opelxpi 4786	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ⟨𝐴, 𝐵⟩ ∈ (ℝ^ × ℝ^*))
3		df-le 8330	. . . . . . 7 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
4	3	eleq2i 2301	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ⟨𝐴, 𝐵⟩ ∈ ((ℝ^* × ℝ^*) ∖ ^◡ < ))
5		eldif 3223	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ((ℝ^* × ℝ^) ∖ ^◡ < ) ↔ (⟨𝐴, 𝐵⟩ ∈ (ℝ^ × ℝ^*) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
6	4, 5	bitri 184	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ (⟨𝐴, 𝐵⟩ ∈ (ℝ^* × ℝ^*) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
7	6	baib 927	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (ℝ^* × ℝ^*) → (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
8	2, 7	syl 14	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
9	1, 8	bitrid 192	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
10		df-br 4115	. . . 4 ⊢ (𝐵 < 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ < )
11		opelcnvg 4940	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (⟨𝐴, 𝐵⟩ ∈ ^◡ < ↔ ⟨𝐵, 𝐴⟩ ∈ < ))
12	10, 11	bitr4id 199	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
13	12	notbid 673	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (¬ 𝐵 < 𝐴 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
14	9, 13	bitr4d 191	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2205 ∖ cdif 3211 ⟨cop 3697 class class class wbr 4114 × cxp 4752 ^◡ccnv 4753 ℝ^*cxr 8323 < clt 8324 ≤ cle 8325

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ral 2527 df-rex 2528 df-v 2817 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-br 4115 df-opab 4177 df-xp 4760 df-cnv 4762 df-le 8330

This theorem is referenced by: lenlt 8365 pnfge 10141 mnfle 10144 xrltle 10150 xrleid 10152 xnn0dcle 10154 xrletri3 10156 xrlelttr 10158 xrltletr 10159 xrletr 10160 xgepnf 10168 xleneg 10189 xltadd1 10228 xsubge0 10233 xleaddadd 10239 iccid 10277 icc0r 10278 icodisj 10344 ioodisj 10345 ioo0 10643 ico0 10645 ioc0 10646 leisorel 11234 xrmaxleim 11954 xrmaxiflemval 11960 xrmaxlesup 11969 xrmaxaddlem 11970 xrminmax 11975 pcadd 13063 bldisj 15392 bdxmet 15492 bdbl 15494

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