Theorem ssiin 4017

Description: Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
ssiin	⊢ (C ⊆ ∩x ∈ A B ↔ ∀x ∈ A C ⊆ B)

Distinct variable group:   x,C

Allowed substitution hints:   A(x)   B(x)

Proof of Theorem ssiin

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲxC
2	1	ssiinf 4016	1 ⊢ (C ⊆ ∩x ∈ A B ↔ ∀x ∈ A C ⊆ B)

Syntax hints:   ↔ wb 176  ∀wral 2615   ⊆ wss 3258  ∩ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iin 3973

This theorem is referenced by: (None)