MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supcl Structured version   Unicode version

Theorem supcl 7909
Description: A supremum belongs to its base class (closure law). See also supub 7910 and suplub 7911. (Contributed by NM, 12-Oct-2004.)
Hypotheses
Ref Expression
supmo.1  |-  ( ph  ->  R  Or  A )
supcl.2  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
Assertion
Ref Expression
supcl  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
Distinct variable groups:    x, y,
z, A    x, R, y, z    x, B, y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem supcl
StepHypRef Expression
1 supmo.1 . . 3  |-  ( ph  ->  R  Or  A )
21supval2 7906 . 2  |-  ( ph  ->  sup ( B ,  A ,  R )  =  ( iota_ x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  (
y R x  ->  E. z  e.  B  y R z ) ) ) )
3 supcl.2 . . . 4  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
41, 3supeu 7905 . . 3  |-  ( ph  ->  E! x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
5 riotacl 6246 . . 3  |-  ( E! x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) )  ->  ( iota_ x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  (
y R x  ->  E. z  e.  B  y R z ) ) )  e.  A )
64, 5syl 16 . 2  |-  ( ph  ->  ( iota_ x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )  e.  A )
72, 6eqeltrd 2542 1  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    e. wcel 1823   A.wral 2804   E.wrex 2805   E!wreu 2806   class class class wbr 4439    Or wor 4788   iota_crio 6231   supcsup 7892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-po 4789  df-so 4790  df-iota 5534  df-riota 6232  df-sup 7893
This theorem is referenced by:  suplub2  7912  supmaxOLD  7917  supiso  7925  suprcl  10498  infmsup  10516  supxrcl  11509  infmxrcl  11511  dgrcl  22796  supssd  27756  xrsupssd  27810  xrge0infssd  27812  esum2d  28322  omsf  28504  oddpwdc  28557  wzel  29620  wsuccl  29623  supclt  30469
  Copyright terms: Public domain W3C validator