Theorem 9re 9393

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 9372	. 2 |- 9 = ( 8 + 1 )
2		8re 9391	. . 3 |- 8 e. RR
3		1re 8325	. . 3 |- 1 e. RR
4	2, 3	readdcli 8339	. 2 |- ( 8 + 1 ) e. RR
5	1, 4	eqeltri 2311	1 |- 9 e. RR

This proof depends on syntax axioms:   e. wcel 2209  (class class class)co 6085   RRcr 8178   1c1 8180   + caddc 8182   8c8 9363   9c9 9364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-2 9365  df-3 9366  df-4 9367  df-5 9368  df-6 9369  df-7 9370  df-8 9371  df-9 9372

This theorem is used by:  9cn  9394  7lt9  9507  6lt9  9508  5lt9  9509  4lt9  9510  3lt9  9511  2lt9  9512  1lt9  9513  9lt10  9916  8lt10  9917  0.999...  12304  cos2bnd  12543  sincos2sgn  12549  slotsdifplendx  13613  dsndxntsetndx  13627  unifndxntsetndx  13634  setsmsdsg  15630  2logb9irr  16126  2logb9irrap  16132  log2tlbndlog2  16139  bposlem4  16212  bposlem5  16213