Theorem intss 3920

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 1482	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)))
3		vex 2779	. . . . 5 ⊢ 𝑥 ∈ V
4	3	elint 3905	. . . 4 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦))
5	3	elint 3905	. . . 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 1898	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ∀𝑥(𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
8		ssalel 3189	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
9		ssalel 3189	. 2 ⊢ (∩ 𝐵 ⊆ ∩ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
10	7, 8, 9	3imtr4i 201	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Syntax hints:   → wi 4  ∀wal 1371   ∈ wcel 2178   ⊆ wss 3174  ∩ cint 3899

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-int 3900

This theorem is referenced by:  lspss  14276  clsss  14705