Theorem 9re 9080

Description: The number 9 is real. (Contributed by NM, 27-May-1999.)

Assertion

Ref	Expression
9re	⊢ 9 ∈ ℝ

Proof of Theorem 9re

Step	Hyp	Ref	Expression
1		df-9 9059	. 2 ⊢ 9 = (8 + 1)
2		8re 9078	. . 3 ⊢ 8 ∈ ℝ
3		1re 8028	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 8042	. 2 ⊢ (8 + 1) ∈ ℝ
5	1, 4	eqeltri 2269	1 ⊢ 9 ∈ ℝ

Syntax hints:   ∈ wcel 2167  (class class class)co 5923  ℝcr 7881  1c1 7883   + caddc 7885  8c8 9050  9c9 9051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-1re 7976  ax-addrcl 7979

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-2 9052  df-3 9053  df-4 9054  df-5 9055  df-6 9056  df-7 9057  df-8 9058  df-9 9059

This theorem is referenced by:  9cn  9081  7lt9  9192  6lt9  9193  5lt9  9194  4lt9  9195  3lt9  9196  2lt9  9197  1lt9  9198  9lt10  9590  8lt10  9591  0.999...  11689  cos2bnd  11928  sincos2sgn  11934  slotsdifplendx  12898  dsndxntsetndx  12912  unifndxntsetndx  12919  setsmsdsg  14742  2logb9irr  15233  2logb9irrap  15239