Axiom ax-nul 5250

Description: The Null Set Axiom of ZF set theory. It was derived as axnul 5249 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 2137	. . . 4 wff 𝑦 ∈ 𝑥
4	3	wn 3	. . 3 wff ¬ 𝑦 ∈ 𝑥
5	4, 1	wal 1552	. 2 wff ∀𝑦 ¬ 𝑦 ∈ 𝑥
6	5, 2	wex 1793	1 wff ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

This axiom is referenced by:  0ex  5251  dtruALT2  5321  axpr  5378  axprlem1OLD  5379  axprlem4OLD  5381  axprlem5OLD  5382  exexneq  5396  axsepg2  35391  axsepg3  35392  axsepg4  35394  axnulg  35396  axtcond  36786  eu0  44044