Theorem bj-axempty2 13081

Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 13080. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4049 instead. (New usage is discouraged.)

Assertion

Ref	Expression
bj-axempty2	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-axempty2

Step	Hyp	Ref	Expression
1		bj-axemptylem 13079	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
2		dfnot 1349	. . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 → ⊥))
3	2	albii 1446	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
4	3	exbii 1584	. 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥))
5	1, 4	mpbir 145	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1329  ⊥wfal 1336  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-bd0 13000  ax-bdim 13001  ax-bdn 13004  ax-bdeq 13007  ax-bdsep 13071

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337

This theorem is referenced by: (None)