Theorem bj-axemptylem 16213

Description: Lemma for bj-axempty 16214 and bj-axempty2 16215. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4209 instead. (New usage is discouraged.)

Assertion

Ref	Expression
bj-axemptylem	⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-axemptylem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdfal 16154	. . 3 ⊢ BOUNDED ⊥
2	1	bdsep1 16206	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥))
3		biimp 118	. . . 4 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → (𝑦 ∈ 𝑥 → (𝑦 ∈ 𝑧 ∧ ⊥)))
4		falimd 1410	. . . 4 ⊢ ((𝑦 ∈ 𝑧 ∧ ⊥) → ⊥)
5	3, 4	syl6 33	. . 3 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → (𝑦 ∈ 𝑥 → ⊥))
6	5	alimi 1501	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
7	2, 6	eximii 1648	1 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1393  ⊥wfal 1400  ∃wex 1538

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580  ax-bd0 16134  ax-bdim 16135  ax-bdn 16138  ax-bdeq 16141  ax-bdsep 16205

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401

This theorem is referenced by:  bj-axempty  16214  bj-axempty2  16215