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Theorem spwcl 14339
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 14338 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   dom cdm 4689  (class class class)co 5858   PosetRelcps 14301    sup w cspw 14303
This theorem is referenced by:  supnuf  25629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-undef 6298  df-riota 6304  df-ps 14306  df-spw 14308
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