Theorem bj-axempty 11784

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 3964. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3965 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 11783	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
2		df-ral 2364	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ⊥ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
3	2	exbii 1541	. 2 ⊢ (∃𝑥∀𝑦 ∈ 𝑥 ⊥ ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥))
4	1, 3	mpbir 144	1 ⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥

Syntax hints:   → wi 4  ∀wal 1287  ⊥wfal 1294  ∃wex 1426  ∀wral 2359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-bd0 11704  ax-bdim 11705  ax-bdn 11708  ax-bdeq 11711  ax-bdsep 11775

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-ral 2364

This theorem is referenced by: (None)