xrlenlt - Intuitionistic Logic Explorer

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Theorem xrlenlt 6881

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 3756	. . 3 ⊢ (A ≤ B ↔ ⟨A, B⟩ ∈ ≤ )
2		opelxpi 4319	. . . 4 ⊢ ((A ∈ ℝ^* ∧ B ∈ ℝ^) → ⟨A, B⟩ ∈ (ℝ^ × ℝ^*))
3		df-le 6863	. . . . . . 7 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
4	3	eleq2i 2101	. . . . . 6 ⊢ (⟨A, B⟩ ∈ ≤ ↔ ⟨A, B⟩ ∈ ((ℝ^* × ℝ^*) ∖ ^◡ < ))
5		eldif 2921	. . . . . 6 ⊢ (⟨A, B⟩ ∈ ((ℝ^* × ℝ^) ∖ ^◡ < ) ↔ (⟨A, B⟩ ∈ (ℝ^ × ℝ^*) ∧ ¬ ⟨A, B⟩ ∈ ^◡ < ))
6	4, 5	bitri 173	. . . . 5 ⊢ (⟨A, B⟩ ∈ ≤ ↔ (⟨A, B⟩ ∈ (ℝ^* × ℝ^*) ∧ ¬ ⟨A, B⟩ ∈ ^◡ < ))
7	6	baib 827	. . . 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	syl5bb 181	. 2 ⊢ ((A ∈ ℝ^* ∧ B ∈ ℝ^*) → (A ≤ B ↔ ¬ ⟨A, B⟩ ∈ ^◡ < ))
10		opelcnvg 4458	. . . 4 ⊢ ((A ∈ ℝ^* ∧ B ∈ ℝ^*) → (⟨A, B⟩ ∈ ^◡ < ↔ ⟨B, A⟩ ∈ < ))
11		df-br 3756	. . . 4 ⊢ (B < A ↔ ⟨B, A⟩ ∈ < )
12	10, 11	syl6rbbr 188	. . 3 ⊢ ((A ∈ ℝ^* ∧ B ∈ ℝ^*) → (B < A ↔ ⟨A, B⟩ ∈ ^◡ < ))
13	12	notbid 591	. 2 ⊢ ((A ∈ ℝ^* ∧ B ∈ ℝ^*) → (¬ B < A ↔ ¬ ⟨A, B⟩ ∈ ^◡ < ))
14	9, 13	bitr4d 180	1 ⊢ ((A ∈ ℝ^* ∧ B ∈ ℝ^*) → (A ≤ B ↔ ¬ B < A))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1390 ∖ cdif 2908 ⟨cop 3370 class class class wbr 3755 × cxp 4286 ^◡ccnv 4287 ℝ^*cxr 6856 < clt 6857 ≤ cle 6858

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935

This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-cnv 4296 df-le 6863

This theorem is referenced by: lenlt 6891 pnfge 8480 mnfle 8483 xrltle 8489 xrleid 8490 xrletri3 8491 xrlelttr 8492 xrltletr 8493 xrletr 8494 xleneg 8520 iccid 8564 icc0r 8565 icodisj 8630 ioodisj 8631

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