Theorem clos1base 5879

Description: The initial set of a closure is a subset of the closure. Theorem IX.5.13 of [Rosser] p. 246. (Contributed by SF, 13-Feb-2015.)

Hypothesis

Ref	Expression
clos1base.1	⊢ C = Clos1 (S, R)

Assertion

Ref	Expression
clos1base	⊢ S ⊆ C

Proof of Theorem clos1base

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssmin 3946	. 2 ⊢ S ⊆ ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)}
2		clos1base.1	. . 3 ⊢ C = Clos1 (S, R)
3		df-clos1 5874	. . 3 ⊢ Clos1 (S, R) = ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)}
4	2, 3	eqtr2i 2374	. 2 ⊢ ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)} = C
5	1, 4	sseqtri 3304	1 ⊢ S ⊆ C

Syntax hints:   ∧ wa 358   = wceq 1642  {cab 2339   ⊆ wss 3258  ∩cint 3927   “ cima 4723   Clos1 cclos1 5873

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  df-clos1 5874

This theorem is referenced by:  clos1induct  5881  clos1basesuc  5883  clos1nrel  5887  sbthlem1  6204  spacid  6286