Theorem bj-axun2 16513

Description: axun2 4532 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 16419	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝑤
2	1	ax-bdex 16417	. . 3 ⊢ BOUNDED ∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤
3		df-rex 2516	. . . 4 ⊢ (∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ 𝑧 ∈ 𝑤))
4		exancom 1656	. . . 4 ⊢ (∃𝑤(𝑤 ∈ 𝑥 ∧ 𝑧 ∈ 𝑤) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
5	3, 4	bitri 184	. . 3 ⊢ (∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
6	2, 5	bd0 16422	. 2 ⊢ BOUNDED ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
7		ax-un 4530	. 2 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
8	6, 7	bdbm1.3ii 16489	1 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1395  ∃wex 1540  ∃wrex 2511

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-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-14 2205  ax-un 4530  ax-bd0 16411  ax-bdex 16417  ax-bdel 16419  ax-bdsep 16482

This theorem depends on definitions:  df-bi 117  df-rex 2516

This theorem is referenced by: (None)