Theorem bj-axun2 13950

Description: axun2 4420 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axun2	⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Proof of Theorem bj-axun2

Step	Hyp	Ref	Expression
1		ax-bdel 13856	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝑤
2	1	ax-bdex 13854	. . 3 ⊢ BOUNDED ∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤
3		df-rex 2454	. . . 4 ⊢ (∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ 𝑧 ∈ 𝑤))
4		exancom 1601	. . . 4 ⊢ (∃𝑤(𝑤 ∈ 𝑥 ∧ 𝑧 ∈ 𝑤) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
5	3, 4	bitri 183	. . 3 ⊢ (∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
6	2, 5	bd0 13859	. 2 ⊢ BOUNDED ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
7		ax-un 4418	. 2 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
8	6, 7	bdbm1.3ii 13926	1 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∀wal 1346  ∃wex 1485  ∃wrex 2449

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-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-14 2144  ax-un 4418  ax-bd0 13848  ax-bdex 13854  ax-bdel 13856  ax-bdsep 13919

This theorem depends on definitions:  df-bi 116  df-rex 2454

This theorem is referenced by: (None)