Theorem intss 3709

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 75	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ((𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)))
2	1	al2imi 1392	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)))
3		vex 2622	. . . . 5 ⊢ 𝑥 ∈ V
4	3	elint 3694	. . . 4 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦))
5	3	elint 3694	. . . 4 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))
6	2, 4, 5	3imtr4g 203	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
7	6	alrimiv 1802	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ∀𝑥(𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
8		dfss2 3014	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
9		dfss2 3014	. 2 ⊢ (∩ 𝐵 ⊆ ∩ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
10	7, 8, 9	3imtr4i 199	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Syntax hints:   → wi 4  ∀wal 1287   ∈ wcel 1438   ⊆ wss 2999  ∩ cint 3688

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-int 3689

This theorem is referenced by: (None)