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Theorem spwcl 14355
Description: Closure of a supremum. (Contributed by NM, 15-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
Hypotheses
Ref Expression
spwex.1  |-  X  =  dom  R
spwex.2  |-  ( ph  <->  ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )
Assertion
Ref Expression
spwcl  |-  ( ( R  e.  PosetRel  /\  A  e.  V  /\  E. x  e.  X  ph )  -> 
( R  sup w  A )  e.  X
)
Distinct variable groups:    x, y,
z, A    x, R, y, z    x, X, y
Allowed substitution hints:    ph( x, y, z)    V( x, y, z)    X( z)

Proof of Theorem spwcl
StepHypRef Expression
1 spwex.1 . . 3  |-  X  =  dom  R
2 spwex.2 . . 3  |-  ( ph  <->  ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )
31, 2spwex 14354 . 2  |-  ( ( R  e.  PosetRel  /\  A  e.  V )  ->  ( E. x  e.  X  ph  <->  ( R  sup w  A
)  e.  X ) )
43biimp3a 1281 1  |-  ( ( R  e.  PosetRel  /\  A  e.  V  /\  E. x  e.  X  ph )  -> 
( R  sup w  A )  e.  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   class class class wbr 4039   dom cdm 4705  (class class class)co 5874   PosetRelcps 14317    sup w cspw 14319
This theorem is referenced by:  supnuf  25732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-undef 6314  df-riota 6320  df-ps 14322  df-spw 14324
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