Definition df-wun 9524

Description: The class of all weak universes. A weak universe is a nonempty transitive class closed under union, pairing, and powerset. The advantage of weak universes over Grothendieck universes is that one can prove that every set is contained in a weak universe in ZF (see uniwun 9562) whereas the analogue for Grothendieck universes requires ax-groth 9645 (see grothtsk 9657). (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
df-wun	|- WUni = { u | ( Tr u /\ u =/= (/) /\ A. x e. u ( U. x e. u /\ ~P x e. u /\ A. y e. u { x , y } e. u ) ) }

Distinct variable group:   x, y, u

Detailed syntax breakdown of Definition df-wun

Step	Hyp	Ref	Expression
1		cwun 9522	. 2 class WUni
2		vu	. . . . . 6 setvar u
3	2	cv 1482	. . . . 5 class u
4	3	wtr 4752	. . . 4 wff Tr u
5		c0 3915	. . . . 5 class (/)
6	3, 5	wne 2794	. . . 4 wff u =/= (/)
7		vx	. . . . . . . . 9 setvar x
8	7	cv 1482	. . . . . . . 8 class x
9	8	cuni 4436	. . . . . . 7 class U. x
10	9, 3	wcel 1990	. . . . . 6 wff U. x e. u
11	8	cpw 4158	. . . . . . 7 class ~P x
12	11, 3	wcel 1990	. . . . . 6 wff ~P x e. u
13		vy	. . . . . . . . . 10 setvar y
14	13	cv 1482	. . . . . . . . 9 class y
15	8, 14	cpr 4179	. . . . . . . 8 class { x , y }
16	15, 3	wcel 1990	. . . . . . 7 wff { x , y } e. u
17	16, 13, 3	wral 2912	. . . . . 6 wff A. y e. u { x , y } e. u
18	10, 12, 17	w3a 1037	. . . . 5 wff ( U. x e. u /\ ~P x e. u /\ A. y e. u { x , y } e. u )
19	18, 7, 3	wral 2912	. . . 4 wff A. x e. u ( U. x e. u /\ ~P x e. u /\ A. y e. u { x , y } e. u )
20	4, 6, 19	w3a 1037	. . 3 wff ( Tr u /\ u =/= (/) /\ A. x e. u ( U. x e. u /\ ~P x e. u /\ A. y e. u { x , y } e. u ) )
21	20, 2	cab 2608	. 2 class { u | ( Tr u /\ u =/= (/) /\ A. x e. u ( U. x e. u /\ ~P x e. u /\ A. y e. u { x , y } e. u ) ) }
22	1, 21	wceq 1483	1 wff WUni = { u | ( Tr u /\ u =/= (/) /\ A. x e. u ( U. x e. u /\ ~P x e. u /\ A. y e. u { x , y } e. u ) ) }

This definition is referenced by:  iswun  9526