Theorem 8re 9389

Description: The number 8 is real. (Contributed by NM, 27-May-1999.)

Assertion

Ref	Expression
8re	⊢ 8 ∈ ℝ

Proof of Theorem 8re

Step	Hyp	Ref	Expression
1		df-8 9369	. 2 ⊢ 8 = (7 + 1)
2		7re 9387	. . 3 ⊢ 7 ∈ ℝ
3		1re 8325	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 8339	. 2 ⊢ (7 + 1) ∈ ℝ
5	1, 4	eqeltri 2311	1 ⊢ 8 ∈ ℝ

This proof depends on syntax axioms:   ∈ wcel 2209  (class class class)co 6085  ℝcr 8178  1c1 8180   + caddc 8182  7c7 9360  8c8 9361

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

This theorem is used by:  8cn  9390  9re  9391  9pos  9408  6lt8  9496  5lt8  9497  4lt8  9498  3lt8  9499  2lt8  9500  1lt8  9501  8lt9  9502  7lt9  9503  8th4div3  9524  8lt10  9908  7lt10  9909  ef01bndlem  12523  cos2bnd  12527  slotstnscsi  13549  slotsdnscsi  13577  2lgsoddprmlem1  16224  2lgsoddprmlem2  16225  2lgsoddprmlem3a  16226  2lgsoddprmlem3b  16227  2lgsoddprmlem3c  16228  2lgsoddprmlem3d  16229