Theorem 9re 8800

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 8779	. 2 ⊢ 9 = (8 + 1)
2		8re 8798	. . 3 ⊢ 8 ∈ ℝ
3		1re 7758	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 7772	. 2 ⊢ (8 + 1) ∈ ℝ
5	1, 4	eqeltri 2210	1 ⊢ 9 ∈ ℝ

Syntax hints:   ∈ wcel 1480  (class class class)co 5767  ℝcr 7612  1c1 7614   + caddc 7616  8c8 8770  9c9 8771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119  ax-1re 7707  ax-addrcl 7710

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133  df-2 8772  df-3 8773  df-4 8774  df-5 8775  df-6 8776  df-7 8777  df-8 8778  df-9 8779

This theorem is referenced by:  9cn  8801  7lt9  8911  6lt9  8912  5lt9  8913  4lt9  8914  3lt9  8915  2lt9  8916  1lt9  8917  9lt10  9305  8lt10  9306  0.999...  11283  cos2bnd  11456  sincos2sgn  11461  setsmsdsg  12638