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Mirrors > Home > MPE Home > Th. List > ax-nul | Structured version Visualization version GIF version |
Description: The Null Set Axiom of ZF set theory. It was derived as axnul 4716 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.) |
Ref | Expression |
---|---|
ax-nul | ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vy | . . . . 5 setvar 𝑦 | |
2 | vx | . . . . 5 setvar 𝑥 | |
3 | 1, 2 | wel 1978 | . . . 4 wff 𝑦 ∈ 𝑥 |
4 | 3 | wn 3 | . . 3 wff ¬ 𝑦 ∈ 𝑥 |
5 | 4, 1 | wal 1473 | . 2 wff ∀𝑦 ¬ 𝑦 ∈ 𝑥 |
6 | 5, 2 | wex 1695 | 1 wff ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff setvar class |
This axiom is referenced by: 0ex 4718 dtru 4783 bj-dtru 31985 |
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