Theorem ssin 3429

Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
ssin	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶))

Proof of Theorem ssin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3390	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
2	1	imbi2i 226	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
3	2	albii 1518	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
4		jcab 607	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
5	4	albii 1518	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
6		19.26 1529	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
7	3, 5, 6	3bitrri 207	. 2 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)))
8		ssalel 3215	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
9		ssalel 3215	. . 3 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
10	8, 9	anbi12i 460	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
11		ssalel 3215	. 2 ⊢ (𝐴 ⊆ (𝐵 ∩ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)))
12	7, 10, 11	3bitr4i 212	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1395   ∈ wcel 2202   ∩ cin 3199   ⊆ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213

This theorem is referenced by:  ssini  3430  ssind  3431  uneqin  3458  trin  4197  pwin  4379  peano5  4696  fin  5523  tgval  13350  eltg3i  14786  innei  14893  cnptoprest2  14970