Theorem 9re 9391

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 9370	. 2 |- 9 = ( 8 + 1 )
2		8re 9389	. . 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 9361   9c9 9362

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 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370

This theorem is used by:  9cn  9392  7lt9  9503  6lt9  9504  5lt9  9505  4lt9  9506  3lt9  9507  2lt9  9508  1lt9  9509  9lt10  9907  8lt10  9908  0.999...  12288  cos2bnd  12527  sincos2sgn  12533  slotsdifplendx  13564  dsndxntsetndx  13578  unifndxntsetndx  13585  setsmsdsg  15581  2logb9irr  16073  2logb9irrap  16079  log2tlbndlog2  16082