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Theorem nfsup 6666
Description: Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
Hypotheses
Ref Expression
nfsup.1  |-  F/_ x A
nfsup.2  |-  F/_ x B
nfsup.3  |-  F/_ x R
Assertion
Ref Expression
nfsup  |-  F/_ x sup ( A ,  B ,  R )

Proof of Theorem nfsup
Dummy variables  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sup 6658 . 2  |-  sup ( A ,  B ,  R )  =  U. { u  e.  B  |  ( A. v  e.  A  -.  u R v  /\  A. v  e.  B  (
v R u  ->  E. w  e.  A  v R w ) ) }
2 nfsup.1 . . . . . 6  |-  F/_ x A
3 nfcv 2228 . . . . . . . 8  |-  F/_ x u
4 nfsup.3 . . . . . . . 8  |-  F/_ x R
5 nfcv 2228 . . . . . . . 8  |-  F/_ x
v
63, 4, 5nfbr 3881 . . . . . . 7  |-  F/ x  u R v
76nfn 1593 . . . . . 6  |-  F/ x  -.  u R v
82, 7nfralya 2416 . . . . 5  |-  F/ x A. v  e.  A  -.  u R v
9 nfsup.2 . . . . . 6  |-  F/_ x B
105, 4, 3nfbr 3881 . . . . . . 7  |-  F/ x  v R u
11 nfcv 2228 . . . . . . . . 9  |-  F/_ x w
125, 4, 11nfbr 3881 . . . . . . . 8  |-  F/ x  v R w
132, 12nfrexya 2417 . . . . . . 7  |-  F/ x E. w  e.  A  v R w
1410, 13nfim 1509 . . . . . 6  |-  F/ x
( v R u  ->  E. w  e.  A  v R w )
159, 14nfralya 2416 . . . . 5  |-  F/ x A. v  e.  B  ( v R u  ->  E. w  e.  A  v R w )
168, 15nfan 1502 . . . 4  |-  F/ x
( A. v  e.  A  -.  u R v  /\  A. v  e.  B  ( v R u  ->  E. w  e.  A  v R w ) )
1716, 9nfrabxy 2547 . . 3  |-  F/_ x { u  e.  B  |  ( A. v  e.  A  -.  u R v  /\  A. v  e.  B  (
v R u  ->  E. w  e.  A  v R w ) ) }
1817nfuni 3654 . 2  |-  F/_ x U. { u  e.  B  |  ( A. v  e.  A  -.  u R v  /\  A. v  e.  B  (
v R u  ->  E. w  e.  A  v R w ) ) }
191, 18nfcxfr 2225 1  |-  F/_ x sup ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   F/_wnfc 2215   A.wral 2359   E.wrex 2360   {crab 2363   U.cuni 3648   class class class wbr 3837   supcsup 6656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-sup 6658
This theorem is referenced by:  nfinf  6691  infssuzcldc  11040
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