Theorem ssintub 3945

Description: Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)

Assertion

Ref	Expression
ssintub	⊢ A ⊆ ∩{x ∈ B ∣ A ⊆ x}

Distinct variable groups:   x,A   x,B

Proof of Theorem ssintub

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 3943	. 2 ⊢ (A ⊆ ∩{x ∈ B ∣ A ⊆ x} ↔ ∀y ∈ {x ∈ B ∣ A ⊆ x}A ⊆ y)
2		sseq2 3294	. . . 4 ⊢ (x = y → (A ⊆ x ↔ A ⊆ y))
3	2	elrab 2995	. . 3 ⊢ (y ∈ {x ∈ B ∣ A ⊆ x} ↔ (y ∈ B ∧ A ⊆ y))
4	3	simprbi 450	. 2 ⊢ (y ∈ {x ∈ B ∣ A ⊆ x} → A ⊆ y)
5	1, 4	mprgbir 2685	1 ⊢ A ⊆ ∩{x ∈ B ∣ A ⊆ x}

Syntax hints:   ∈ wcel 1710  {crab 2619   ⊆ 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-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is referenced by:  intmin  3947