Theorem sn-00id

Description: 00id proven without ax-mulcom but using ax-1ne0 . (Though note that the current version of 00id can be changed to avoid ax-icn , ax-addcl , ax-mulcl , ax-i2m1 , ax-cnre . Most of this is by using 0cnALT3 instead of 0cn ). (Contributed by SN, 25-Dec-2023) (Proof modification is discouraged.)

	Ref	Expression
Assertion	sn-00id	⊢ ( 0 + 0 ) = 0

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		resubadd	⊢ ( ( 0 ∈ ℝ ∧ 0 ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 0 −ℝ 0 ) = 0 ↔ ( 0 + 0 ) = 0 ) )
3	1 1 1 2	mp3an	⊢ ( ( 0 −ℝ 0 ) = 0 ↔ ( 0 + 0 ) = 0 )
4		df-ne	⊢ ( ( 0 −ℝ 0 ) ≠ 0 ↔ ¬ ( 0 −ℝ 0 ) = 0 )
5		sn-00idlem2	⊢ ( ( 0 −ℝ 0 ) ≠ 0 → ( 0 −ℝ 0 ) = 1 )
6		sn-00idlem3	⊢ ( ( 0 −ℝ 0 ) = 1 → ( 0 + 0 ) = 0 )
7	5 6	syl	⊢ ( ( 0 −ℝ 0 ) ≠ 0 → ( 0 + 0 ) = 0 )
8	4 7	sylbir	⊢ ( ¬ ( 0 −ℝ 0 ) = 0 → ( 0 + 0 ) = 0 )
9	3 8	sylnbir	⊢ ( ¬ ( 0 + 0 ) = 0 → ( 0 + 0 ) = 0 )
10	9	pm2.18i	⊢ ( 0 + 0 ) = 0