Theorem renepnfd 11026

Description: No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)

Hypothesis

Ref	Expression
rexrd.1	⊢ (𝜑 → 𝐴 ∈ ℝ)

Assertion

Ref	Expression
renepnfd	⊢ (𝜑 → 𝐴 ≠ +∞)

Proof of Theorem renepnfd

Step	Hyp	Ref	Expression
1		rexrd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		renepnf 11023	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2943  ℝcr 10870  +∞cpnf 11006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-resscn 10928

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-nel 3050  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564  df-uni 4840  df-pnf 11011

This theorem is referenced by:  xaddnepnf  12971  dvfsumrlimge0  25194  dvfsumrlim  25195  dvfsumrlim2  25196  logno1  25791  limsupresico  43241  limsupvaluz2  43279  supcnvlimsup  43281  liminfresico  43312  xlimliminflimsup  43403  smflimsuplem2  44354  smflimsuplem5  44357