Theorem bj-axempty 10013

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 non-empty universe. See axnul 3882. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 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 10012	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
2		df-ral 2311	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ⊥ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
3	2	exbii 1496	. 2 ⊢ (∃𝑥∀𝑦 ∈ 𝑥 ⊥ ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥))
4	1, 3	mpbir 134	1 ⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥

Syntax hints:   → wi 4  ∀wal 1241  ⊥wfal 1248  ∃wex 1381  ∀wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-bd0 9933  ax-bdim 9934  ax-bdn 9937  ax-bdeq 9940  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-ral 2311

This theorem is referenced by: (None)