Axiom ax-rnegex 6752

Description: Existence of negative of real number. Axiom for real and complex numbers, justified by theorem axrnegex 6723. (Contributed by Eric Schmidt, 21-May-2007.)

Assertion

Ref	Expression
ax-rnegex	⊢ (A ∈ ℝ → ∃x ∈ ℝ (A + x) = 0)

Distinct variable group:   x,A

Detailed syntax breakdown of Axiom ax-rnegex

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cr 6670	. . 3 class ℝ
3	1, 2	wcel 1390	. 2 wff A ∈ ℝ
4		vx	. . . . . 6 setvar x
5	4	cv 1241	. . . . 5 class x
6		caddc 6674	. . . . 5 class +
7	1, 5, 6	co 5455	. . . 4 class (A + x)
8		cc0 6671	. . . 4 class 0
9	7, 8	wceq 1242	. . 3 wff (A + x) = 0
10	9, 4, 2	wrex 2301	. 2 wff ∃x ∈ ℝ (A + x) = 0
11	3, 10	wi 4	1 wff (A ∈ ℝ → ∃x ∈ ℝ (A + x) = 0)

This axiom is referenced by:  0re  6785  readdcan  6910  cnegexlem1  6943  cnegexlem2  6944  cnegexlem3  6945  cnegex  6946  renegcl  7028  ltadd2  7172