Axiom ax-xp 4080

Description: State the axiom of cross product. This axiom guarantees the existence of the (Kuratowski) cross product of V with x. Axiom P5 of [Hailperin] p. 10. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
ax-xp	⊢ ∃y∀z(z ∈ y ↔ ∃w∃t(z = ⟪w, t⟫ ∧ t ∈ x))

Distinct variable group:   x,y,z,w,t

Detailed syntax breakdown of Axiom ax-xp

Step	Hyp	Ref	Expression
1		vz	. . . . 5 setvar z
2		vy	. . . . 5 setvar y
3	1, 2	wel 1711	. . . 4 wff z ∈ y
4	1	cv 1641	. . . . . . . 8 class z
5		vw	. . . . . . . . . 10 setvar w
6	5	cv 1641	. . . . . . . . 9 class w
7		vt	. . . . . . . . . 10 setvar t
8	7	cv 1641	. . . . . . . . 9 class t
9	6, 8	copk 4058	. . . . . . . 8 class ⟪w, t⟫
10	4, 9	wceq 1642	. . . . . . 7 wff z = ⟪w, t⟫
11		vx	. . . . . . . 8 setvar x
12	7, 11	wel 1711	. . . . . . 7 wff t ∈ x
13	10, 12	wa 358	. . . . . 6 wff (z = ⟪w, t⟫ ∧ t ∈ x)
14	13, 7	wex 1541	. . . . 5 wff ∃t(z = ⟪w, t⟫ ∧ t ∈ x)
15	14, 5	wex 1541	. . . 4 wff ∃w∃t(z = ⟪w, t⟫ ∧ t ∈ x)
16	3, 15	wb 176	. . 3 wff (z ∈ y ↔ ∃w∃t(z = ⟪w, t⟫ ∧ t ∈ x))
17	16, 1	wal 1540	. 2 wff ∀z(z ∈ y ↔ ∃w∃t(z = ⟪w, t⟫ ∧ t ∈ x))
18	17, 2	wex 1541	1 wff ∃y∀z(z ∈ y ↔ ∃w∃t(z = ⟪w, t⟫ ∧ t ∈ x))

This axiom is used by:  axxpprim  4091  xpkvexg  4286