Theorem xrlenltt 5513

Description: 'Less than or equal to' expressed in terms of 'less than', for extended reals.

Assertion

Ref	Expression
xrlenltt	⊢ ((A ∈ ℝ* ⋀ B ∈ ℝ*) → (A ≤ B ↔ ¬ B < A))

Proof of Theorem xrlenltt

Step	Hyp	Ref	Expression
1		opelxpi 3223	. . . 4 ⊢ ((A ∈ ℝ* ⋀ B ∈ ℝ*) → ⟨A, B⟩ ∈ (ℝ* × ℝ*))
2		df-le 5503	. . . . . . 7 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < )
3	2	eleq2i 1541	. . . . . 6 ⊢ (⟨A, B⟩ ∈ ≤ ↔ ⟨A, B⟩ ∈ ((ℝ* × ℝ*) ∖ ◡ < ))
4		eldif 2060	. . . . . 6 ⊢ (⟨A, B⟩ ∈ ((ℝ* × ℝ*) ∖ ◡ < ) ↔ (⟨A, B⟩ ∈ (ℝ* × ℝ*) ⋀ ¬ ⟨A, B⟩ ∈ ◡ < ))
5	3, 4	bitr 173	. . . . 5 ⊢ (⟨A, B⟩ ∈ ≤ ↔ (⟨A, B⟩ ∈ (ℝ* × ℝ*) ⋀ ¬ ⟨A, B⟩ ∈ ◡ < ))
6	5	baib 687	. . . 4 ⊢ (⟨A, B⟩ ∈ (ℝ* × ℝ*) → (⟨A, B⟩ ∈ ≤ ↔ ¬ ⟨A, B⟩ ∈ ◡ < ))
7	1, 6	syl 10	. . 3 ⊢ ((A ∈ ℝ* ⋀ B ∈ ℝ*) → (⟨A, B⟩ ∈ ≤ ↔ ¬ ⟨A, B⟩ ∈ ◡ < ))
8		df-br 2625	. . 3 ⊢ (A ≤ B ↔ ⟨A, B⟩ ∈ ≤ )
9	7, 8	syl5bb 534	. 2 ⊢ ((A ∈ ℝ* ⋀ B ∈ ℝ*) → (A ≤ B ↔ ¬ ⟨A, B⟩ ∈ ◡ < ))
10		opelcnvg 3302	. . . 4 ⊢ ((A ∈ ℝ* ⋀ B ∈ ℝ*) → (⟨A, B⟩ ∈ ◡ < ↔ ⟨B, A⟩ ∈ < ))
11		df-br 2625	. . . 4 ⊢ (B < A ↔ ⟨B, A⟩ ∈ < )
12	10, 11	syl6rbbr 541	. . 3 ⊢ ((A ∈ ℝ* ⋀ B ∈ ℝ*) → (B < A ↔ ⟨A, B⟩ ∈ ◡ < ))
13	12	negbid 613	. 2 ⊢ ((A ∈ ℝ* ⋀ B ∈ ℝ*) → (¬ B < A ↔ ¬ ⟨A, B⟩ ∈ ◡ < ))
14	9, 13	bitr4d 533	1 ⊢ ((A ∈ ℝ* ⋀ B ∈ ℝ*) → (A ≤ B ↔ ¬ B < A))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   ∈ wcel 960   ∖ cdif 2047  ⟨cop 2415   class class class wbr 2624   × cxp 3174  ^◡ccnv 3175   ≤ cle 5307  ℝ^*cxr 5497   < clt 5498

This theorem is referenced by:  xrltnlet 5514  lenltt 5522  pnfget 5560  mnflet 5561  xrleloet 5569  supxr2 6084  supxrbnd 6093  supxrbnd1 6097  supxrbnd2 6098  supxrub 6100  supxrleub 6101  ioon0t 6370  nmlnogt0 8453  iintlem1 10603

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-le 5503

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