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Metamath Proof Explorer

Theorem xrre3

Description: A way of proving that an extended real is real. (Contributed by FL, 29-May-2014)

Ref Expression
Assertion xrre3 ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐵 ≤ 𝐴 ∧ 𝐴 < +∞ ) ) → 𝐴 ∈ ℝ )

Proof

Step Hyp Ref Expression
1 mnflt ⊢ ( 𝐵 ∈ ℝ → -∞ < 𝐵 )
2 1 adantl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → -∞ < 𝐵 )
3 mnfxr ⊢ -∞ ∈ ℝ*
4 rexr ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* )
5 4 adantl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → 𝐵 ∈ ℝ* )
6 simpl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → 𝐴 ∈ ℝ* )
7 xrltletr ⊢ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( ( -∞ < 𝐵 ∧ 𝐵 ≤ 𝐴 ) → -∞ < 𝐴 ) )
8 3 5 6 7 mp3an2i ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( ( -∞ < 𝐵 ∧ 𝐵 ≤ 𝐴 ) → -∞ < 𝐴 ) )
9 2 8 mpand ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( 𝐵 ≤ 𝐴 → -∞ < 𝐴 ) )
10 9 imp ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 ≤ 𝐴 ) → -∞ < 𝐴 )
11 10 adantrr ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐵 ≤ 𝐴 ∧ 𝐴 < +∞ ) ) → -∞ < 𝐴 )
12 simprr ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐵 ≤ 𝐴 ∧ 𝐴 < +∞ ) ) → 𝐴 < +∞ )
13 xrrebnd ⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ∈ ℝ ↔ ( -∞ < 𝐴 ∧ 𝐴 < +∞ ) ) )
14 13 ad2antrr ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐵 ≤ 𝐴 ∧ 𝐴 < +∞ ) ) → ( 𝐴 ∈ ℝ ↔ ( -∞ < 𝐴 ∧ 𝐴 < +∞ ) ) )
15 11 12 14 mpbir2and ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐵 ≤ 𝐴 ∧ 𝐴 < +∞ ) ) → 𝐴 ∈ ℝ )