Theorem bj-axemptylem 10683

Description: Lemma for bj-axempty 10684 and bj-axempty2 10685. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3904 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 10624	. . 3 ⊢ BOUNDED ⊥
2	1	bdsep1 10676	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥))
3		bi1 116	. . . 4 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → (𝑦 ∈ 𝑥 → (𝑦 ∈ 𝑧 ∧ ⊥)))
4		falimd 1299	. . . 4 ⊢ ((𝑦 ∈ 𝑧 ∧ ⊥) → ⊥)
5	3, 4	syl6 33	. . 3 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → (𝑦 ∈ 𝑥 → ⊥))
6	5	alimi 1384	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → ∀𝑦(𝑦 ∈ 𝑥 → ⊥))
7	2, 6	eximii 1533	1 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282  ⊥wfal 1289  ∃wex 1421

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 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-bd0 10604  ax-bdim 10605  ax-bdn 10608  ax-bdeq 10611  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by:  bj-axempty  10684  bj-axempty2  10685