Theorem 9re 9024

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

Assertion

Ref	Expression
9re	|- 9 e. RR

Proof of Theorem 9re

Step	Hyp	Ref	Expression
1		df-9 9003	. 2 |- 9 = ( 8 + 1 )
2		8re 9022	. . 3 |- 8 e. RR
3		1re 7974	. . 3 |- 1 e. RR
4	2, 3	readdcli 7988	. 2 |- ( 8 + 1 ) e. RR
5	1, 4	eqeltri 2262	1 |- 9 e. RR

Syntax hints:   e. wcel 2160  (class class class)co 5891   RRcr 7828   1c1 7830   + caddc 7832   8c8 8994   9c9 8995

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171  ax-1re 7923  ax-addrcl 7926

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185  df-2 8996  df-3 8997  df-4 8998  df-5 8999  df-6 9000  df-7 9001  df-8 9002  df-9 9003

This theorem is referenced by:  9cn  9025  7lt9  9135  6lt9  9136  5lt9  9137  4lt9  9138  3lt9  9139  2lt9  9140  1lt9  9141  9lt10  9532  8lt10  9533  0.999...  11547  cos2bnd  11786  sincos2sgn  11791  slotsdifplendx  12687  dsndxntsetndx  12697  unifndxntsetndx  12704  cnfldstr  13827  setsmsdsg  14377  2logb9irr  14786  2logb9irrap  14792