Axiom ax-nul 5252

Description: The Null Set Axiom of ZF set theory. It was derived as axnul 5251 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 7-Aug-2003.)

Assertion

Ref	Expression
ax-nul	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Axiom ax-nul

Step	Hyp	Ref	Expression
1		vy	. . . . 5 setvar 𝑦
2		vx	. . . . 5 setvar 𝑥
3	1, 2	wel 2115	. . . 4 wff 𝑦 ∈ 𝑥
4	3	wn 3	. . 3 wff ¬ 𝑦 ∈ 𝑥
5	4, 1	wal 1540	. 2 wff ∀𝑦 ¬ 𝑦 ∈ 𝑥
6	5, 2	wex 1781	1 wff ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

This axiom is referenced by:  0ex  5253  dtruALT2  5316  axprlem1  5369  axpr  5373  axprlem4OLD  5375  axprlem5OLD  5376  exexneq  5385  axsepg2  35219  axnulg  35245  eu0  43797