Theorem intss 3636

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 70	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ((𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)))
2	1	al2imi 1347	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)))
3		vex 2560	. . . . 5 ⊢ 𝑥 ∈ V
4	3	elint 3621	. . . 4 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦))
5	3	elint 3621	. . . 4 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))
6	2, 4, 5	3imtr4g 194	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
7	6	alrimiv 1754	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ∀𝑥(𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
8		dfss2 2934	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
9		dfss2 2934	. 2 ⊢ (∩ 𝐵 ⊆ ∩ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴))
10	7, 8, 9	3imtr4i 190	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Syntax hints:   → wi 4  ∀wal 1241   ∈ wcel 1393   ⊆ wss 2917  ∩ cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-int 3616

This theorem is referenced by: (None)