Theorem bj-axempty 13610

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 4101. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4102 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 13609	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
2		df-ral 2447	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ⊥ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
3	2	exbii 1592	. 2 ⊢ (∃𝑥∀𝑦 ∈ 𝑥 ⊥ ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥))
4	1, 3	mpbir 145	1 ⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥

Syntax hints:   → wi 4  ∀wal 1340  ⊥wfal 1347  ∃wex 1479  ∀wral 2442

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 604  ax-in2 605  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-ial 1521  ax-bd0 13530  ax-bdim 13531  ax-bdn 13534  ax-bdeq 13537  ax-bdsep 13601

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-fal 1348  df-ral 2447

This theorem is referenced by: (None)