Theorem ssintab 3944

Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
ssintab	⊢ (A ⊆ ∩{x ∣ φ} ↔ ∀x(φ → A ⊆ x))

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem ssintab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 3943	. 2 ⊢ (A ⊆ ∩{x ∣ φ} ↔ ∀y ∈ {x ∣ φ}A ⊆ y)
2		sseq2 3294	. . 3 ⊢ (y = x → (A ⊆ y ↔ A ⊆ x))
3	2	ralab2 3002	. 2 ⊢ (∀y ∈ {x ∣ φ}A ⊆ y ↔ ∀x(φ → A ⊆ x))
4	1, 3	bitri 240	1 ⊢ (A ⊆ ∩{x ∣ φ} ↔ ∀x(φ → A ⊆ x))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  {cab 2339  ∀wral 2615   ⊆ wss 3258  ∩cint 3927

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-int 3928

This theorem is referenced by:  ssmin  3946  ssintrab  3950  intmin4  3956