Metamath Proof Explorer


Theorem unissb

Description: Relationship involving membership, subset, and union. Exercise 5 of Enderton p. 26 and its converse. (Contributed by NM, 20-Sep-2003) Avoid ax-11 . (Revised by BTernaryTau, 28-Dec-2024)

Ref Expression
Assertion unissb ( 𝐴𝐵 ↔ ∀ 𝑥𝐴 𝑥𝐵 )

Proof

Step Hyp Ref Expression
1 eluni ( 𝑦 𝐴 ↔ ∃ 𝑥 ( 𝑦𝑥𝑥𝐴 ) )
2 1 imbi1i ( ( 𝑦 𝐴𝑦𝐵 ) ↔ ( ∃ 𝑥 ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) )
3 19.23v ( ∀ 𝑥 ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) ↔ ( ∃ 𝑥 ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) )
4 2 3 bitr4i ( ( 𝑦 𝐴𝑦𝐵 ) ↔ ∀ 𝑥 ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) )
5 4 albii ( ∀ 𝑦 ( 𝑦 𝐴𝑦𝐵 ) ↔ ∀ 𝑦𝑥 ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) )
6 elequ1 ( 𝑦 = 𝑧 → ( 𝑦𝑥𝑧𝑥 ) )
7 6 anbi1d ( 𝑦 = 𝑧 → ( ( 𝑦𝑥𝑥𝐴 ) ↔ ( 𝑧𝑥𝑥𝐴 ) ) )
8 eleq1w ( 𝑦 = 𝑧 → ( 𝑦𝐵𝑧𝐵 ) )
9 7 8 imbi12d ( 𝑦 = 𝑧 → ( ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) ↔ ( ( 𝑧𝑥𝑥𝐴 ) → 𝑧𝐵 ) ) )
10 elequ2 ( 𝑥 = 𝑧 → ( 𝑦𝑥𝑦𝑧 ) )
11 eleq1w ( 𝑥 = 𝑧 → ( 𝑥𝐴𝑧𝐴 ) )
12 10 11 anbi12d ( 𝑥 = 𝑧 → ( ( 𝑦𝑥𝑥𝐴 ) ↔ ( 𝑦𝑧𝑧𝐴 ) ) )
13 12 imbi1d ( 𝑥 = 𝑧 → ( ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) ↔ ( ( 𝑦𝑧𝑧𝐴 ) → 𝑦𝐵 ) ) )
14 9 13 alcomw ( ∀ 𝑦𝑥 ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) ↔ ∀ 𝑥𝑦 ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) )
15 19.21v ( ∀ 𝑦 ( 𝑥𝐴 → ( 𝑦𝑥𝑦𝐵 ) ) ↔ ( 𝑥𝐴 → ∀ 𝑦 ( 𝑦𝑥𝑦𝐵 ) ) )
16 impexp ( ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) ↔ ( 𝑦𝑥 → ( 𝑥𝐴𝑦𝐵 ) ) )
17 bi2.04 ( ( 𝑦𝑥 → ( 𝑥𝐴𝑦𝐵 ) ) ↔ ( 𝑥𝐴 → ( 𝑦𝑥𝑦𝐵 ) ) )
18 16 17 bitri ( ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) ↔ ( 𝑥𝐴 → ( 𝑦𝑥𝑦𝐵 ) ) )
19 18 albii ( ∀ 𝑦 ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) ↔ ∀ 𝑦 ( 𝑥𝐴 → ( 𝑦𝑥𝑦𝐵 ) ) )
20 dfss2 ( 𝑥𝐵 ↔ ∀ 𝑦 ( 𝑦𝑥𝑦𝐵 ) )
21 20 imbi2i ( ( 𝑥𝐴𝑥𝐵 ) ↔ ( 𝑥𝐴 → ∀ 𝑦 ( 𝑦𝑥𝑦𝐵 ) ) )
22 15 19 21 3bitr4i ( ∀ 𝑦 ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) ↔ ( 𝑥𝐴𝑥𝐵 ) )
23 22 albii ( ∀ 𝑥𝑦 ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) ↔ ∀ 𝑥 ( 𝑥𝐴𝑥𝐵 ) )
24 14 23 bitri ( ∀ 𝑦𝑥 ( ( 𝑦𝑥𝑥𝐴 ) → 𝑦𝐵 ) ↔ ∀ 𝑥 ( 𝑥𝐴𝑥𝐵 ) )
25 5 24 bitri ( ∀ 𝑦 ( 𝑦 𝐴𝑦𝐵 ) ↔ ∀ 𝑥 ( 𝑥𝐴𝑥𝐵 ) )
26 dfss2 ( 𝐴𝐵 ↔ ∀ 𝑦 ( 𝑦 𝐴𝑦𝐵 ) )
27 df-ral ( ∀ 𝑥𝐴 𝑥𝐵 ↔ ∀ 𝑥 ( 𝑥𝐴𝑥𝐵 ) )
28 25 26 27 3bitr4i ( 𝐴𝐵 ↔ ∀ 𝑥𝐴 𝑥𝐵 )