Theorem bj-axempty2 16590

Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 16589. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4220 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 16588	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
2		dfnot 1416	. . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 → ⊥))
3	2	albii 1519	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
4	3	exbii 1654	. 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥))
5	1, 4	mpbir 146	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1396  ⊥wfal 1403  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583  ax-bd0 16509  ax-bdim 16510  ax-bdn 16513  ax-bdeq 16516  ax-bdsep 16580

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404

This theorem is referenced by: (None)