Theorem bj-axempty 12492

Description: Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a nonempty universe. See axnul 3985. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3986 instead. (New usage is discouraged.)

Assertion

Ref	Expression
bj-axempty	⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-axempty

Step	Hyp	Ref	Expression
1		bj-axemptylem 12491	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
2		df-ral 2375	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ⊥ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
3	2	exbii 1548	. 2 ⊢ (∃𝑥∀𝑦 ∈ 𝑥 ⊥ ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥))
4	1, 3	mpbir 145	1 ⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥

Syntax hints:   → wi 4  ∀wal 1294  ⊥wfal 1301  ∃wex 1433  ∀wral 2370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-ial 1479  ax-bd0 12412  ax-bdim 12413  ax-bdn 12416  ax-bdeq 12419  ax-bdsep 12483

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-ral 2375

This theorem is referenced by: (None)