Theorem intss1 3725

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 2636	. . . 4 ⊢ 𝑥 ∈ V
2	1	elint 3716	. . 3 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦))
3		eleq1 2157	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4		eleq2 2158	. . . . . 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 2720	. . . 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 3045	1 ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4  ∀wal 1294   = wceq 1296   ∈ wcel 1445   ⊆ wss 3013  ∩ cint 3710

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-int 3711

This theorem is referenced by:  intminss  3735  intmin3  3737  intab  3739  int0el  3740  trintssm  3974  inteximm  4006  onnmin  4412  peano5  4441  peano5nnnn  7524  peano5nni  8523  dfuzi  8955  bj-intabssel  12397  bj-intabssel1  12398