Theorem intss1 3846

Description: An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)

Assertion

Ref	Expression
intss1	⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴)

Proof of Theorem intss1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2733	. . . 4 ⊢ 𝑥 ∈ V
2	1	elint 3837	. . 3 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦))
3		eleq1 2233	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4		eleq2 2234	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴))
5	3, 4	imbi12d 233	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) ↔ (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐴)))
6	5	spcgv 2817	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐴)))
7	6	pm2.43a 51	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝐴))
8	2, 7	syl5bi 151	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ 𝐴))
9	8	ssrdv 3153	1 ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4  ∀wal 1346   = wceq 1348   ∈ wcel 2141   ⊆ wss 3121  ∩ cint 3831

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-int 3832

This theorem is referenced by:  intminss  3856  intmin3  3858  intab  3860  int0el  3861  trintssm  4103  inteximm  4135  onnmin  4552  peano5  4582  peano5nnnn  7854  peano5nni  8881  dfuzi  9322  bj-intabssel  13824  bj-intabssel1  13825