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Theorem clos1base 5878
 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.
StepHypRef Expression
1 ssmin 3945 . 2 S {a (S a (Ra) a)}
2 clos1base.1 . . 3 C = Clos1 (S, R)
3 df-clos1 5873 . . 3 Clos1 (S, R) = {a (S a (Ra) a)}
42, 3eqtr2i 2374 . 2 {a (S a (Ra) a)} = C
51, 4sseqtri 3303 1 S C
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642  {cab 2339   ⊆ wss 3257  ∩cint 3926   “ cima 4722   Clos1 cclos1 5872 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927  df-clos1 5873 This theorem is referenced by:  clos1induct  5880  clos1basesuc  5882  clos1nrel  5886  sbthlem1  6203  spacid  6285
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