Theorem 9re 9006

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 8985	. 2 ⊢ 9 = (8 + 1)
2		8re 9004	. . 3 ⊢ 8 ∈ ℝ
3		1re 7956	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 7970	. 2 ⊢ (8 + 1) ∈ ℝ
5	1, 4	eqeltri 2250	1 ⊢ 9 ∈ ℝ

Syntax hints:   ∈ wcel 2148  (class class class)co 5875  ℝcr 7810  1c1 7812   + caddc 7814  8c8 8976  9c9 8977

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-2 8978  df-3 8979  df-4 8980  df-5 8981  df-6 8982  df-7 8983  df-8 8984  df-9 8985

This theorem is referenced by:  9cn  9007  7lt9  9117  6lt9  9118  5lt9  9119  4lt9  9120  3lt9  9121  2lt9  9122  1lt9  9123  9lt10  9514  8lt10  9515  0.999...  11529  cos2bnd  11768  sincos2sgn  11773  slotsdifplendx  12665  dsndxntsetndx  12675  unifndxntsetndx  12682  cnfldstr  13460  setsmsdsg  13983  2logb9irr  14392  2logb9irrap  14398