Theorem bj-axempty 16166

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 4188. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4189 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 16165	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
2		df-ral 2493	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ⊥ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
3	2	exbii 1631	. 2 ⊢ (∃𝑥∀𝑦 ∈ 𝑥 ⊥ ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥))
4	1, 3	mpbir 146	1 ⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥

Syntax hints:   → wi 4  ∀wal 1373  ⊥wfal 1380  ∃wex 1518  ∀wral 2488

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 617  ax-in2 618  ax-5 1473  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-4 1536  ax-ial 1560  ax-bd0 16086  ax-bdim 16087  ax-bdn 16090  ax-bdeq 16093  ax-bdsep 16157

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-fal 1381  df-ral 2493

This theorem is referenced by: (None)