Theorem bj-axemptylem 13927

Description: Lemma for bj-axempty 13928 and bj-axempty2 13929. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4115 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 13868	. . 3 ⊢ BOUNDED ⊥
2	1	bdsep1 13920	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥))
3		biimp 117	. . . 4 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → (𝑦 ∈ 𝑥 → (𝑦 ∈ 𝑧 ∧ ⊥)))
4		falimd 1363	. . . 4 ⊢ ((𝑦 ∈ 𝑧 ∧ ⊥) → ⊥)
5	3, 4	syl6 33	. . 3 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → (𝑦 ∈ 𝑥 → ⊥))
6	5	alimi 1448	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
7	2, 6	eximii 1595	1 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  ⊥wfal 1353  ∃wex 1485

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527  ax-bd0 13848  ax-bdim 13849  ax-bdn 13852  ax-bdeq 13855  ax-bdsep 13919

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354

This theorem is referenced by:  bj-axempty  13928  bj-axempty2  13929