Theorem renepnfd 11387

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 11384	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ≠ wne 2959  ℝcr 11226  +∞cpnf 11367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-resscn 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-nel 3066  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-pw 4562  df-uni 4871  df-pnf 11372

This theorem is used by:  xaddnepnf  13394  dvfsumrlimge0  26380  dvfsumrlim  26381  dvfsumrlim2  26382  logno1  26994  rexmul2  33416  xnn0nn0d  33434  fldextrspundgdvdslem  34382  limsupresico  46915  limsupvaluz2  46953  supcnvlimsup  46955  liminfresico  46986  xlimliminflimsup  47077  smflimsuplem2  48036  smflimsuplem5  48039