Theorem intss 3969

Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)

Assertion

Ref	Expression
intss	⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Proof of Theorem intss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imim1 76	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ((𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)))
2	1	al2imi 1507	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)))
3		vex 2815	. . . . 5 ⊢ 𝑥 ∈ V
4	3	elint 3954	. . . 4 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦))
5	3	elint 3954	. . . 4 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))
6	2, 4, 5	3imtr4g 205	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
7	6	alrimiv 1923	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ∀𝑥(𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
8		ssalel 3225	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
9		ssalel 3225	. 2 ⊢ (∩ 𝐵 ⊆ ∩ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
10	7, 8, 9	3imtr4i 201	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Syntax hints:   → wi 4  ∀wal 1396   ∈ wcel 2203   ⊆ wss 3210  ∩ cint 3948

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-int 3949

This theorem is referenced by:  lspss  14534  clsss  14970