Theorem xrltnlet 5514

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

Assertion

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

Proof of Theorem xrltnlet

Step	Hyp	Ref	Expression
1		xrlenltt 5513	. . 3 ⊢ ((B ∈ ℝ* ⋀ A ∈ ℝ*) → (B ≤ A ↔ ¬ A < B))
2	1	con2bid 528	. 2 ⊢ ((B ∈ ℝ* ⋀ A ∈ ℝ*) → (A < B ↔ ¬ B ≤ A))
3	2	ancoms 438	1 ⊢ ((A ∈ ℝ* ⋀ B ∈ ℝ*) → (A < B ↔ ¬ B ≤ A))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   ∈ wcel 960   class class class wbr 2624   ≤ cle 5307  ℝ^*cxr 5497   < clt 5498

This theorem is referenced by:  xrletrit 5573  ioo0t 6369  cdrci 10480

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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