Theorem bj-axempty2 10685

Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 10684. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3904 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 10683	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
2		dfnot 1302	. . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 → ⊥))
3	2	albii 1399	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
4	3	exbii 1536	. 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥))
5	1, 4	mpbir 144	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1282  ⊥wfal 1289  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-bd0 10604  ax-bdim 10605  ax-bdn 10608  ax-bdeq 10611  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by: (None)