Theorem bj-axempty 15903

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 4173. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4174 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 15902	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
2		df-ral 2490	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ⊥ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
3	2	exbii 1629	. 2 ⊢ (∃𝑥∀𝑦 ∈ 𝑥 ⊥ ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥))
4	1, 3	mpbir 146	1 ⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥

Syntax hints:   → wi 4  ∀wal 1371  ⊥wfal 1378  ∃wex 1516  ∀wral 2485

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 615  ax-in2 616  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558  ax-bd0 15823  ax-bdim 15824  ax-bdn 15827  ax-bdeq 15830  ax-bdsep 15894

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-ral 2490

This theorem is referenced by: (None)