Theorem rintm 3958

Description: Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)

Assertion

Ref	Expression
rintm	⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)

Distinct variable group:   𝑥,𝑋

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem rintm

Step	Hyp	Ref	Expression
1		incom 3314	. 2 ⊢ (𝐴 ∩ ∩ 𝑋) = (∩ 𝑋 ∩ 𝐴)
2		intssuni2m 3848	. . . 4 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴)
3		ssid 3162	. . . . 5 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴
4		sspwuni 3950	. . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴)
5	3, 4	mpbi 144	. . . 4 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴
6	2, 5	sstrdi 3154	. . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ 𝐴)
7		df-ss 3129	. . 3 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (∩ 𝑋 ∩ 𝐴) = ∩ 𝑋)
8	6, 7	sylib 121	. 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (∩ 𝑋 ∩ 𝐴) = ∩ 𝑋)
9	1, 8	syl5eq 2211	1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136   ∩ cin 3115   ⊆ wss 3116  𝒫 cpw 3559  ∪ cuni 3789  ∩ cint 3824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-uni 3790  df-int 3825

This theorem is referenced by: (None)