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Theorem nfsup 6831
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 6823 . 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 2255 . . . . . . . 8  |-  F/_ x u
4 nfsup.3 . . . . . . . 8  |-  F/_ x R
5 nfcv 2255 . . . . . . . 8  |-  F/_ x
v
63, 4, 5nfbr 3939 . . . . . . 7  |-  F/ x  u R v
76nfn 1619 . . . . . 6  |-  F/ x  -.  u R v
82, 7nfralya 2447 . . . . 5  |-  F/ x A. v  e.  A  -.  u R v
9 nfsup.2 . . . . . 6  |-  F/_ x B
105, 4, 3nfbr 3939 . . . . . . 7  |-  F/ x  v R u
11 nfcv 2255 . . . . . . . . 9  |-  F/_ x w
125, 4, 11nfbr 3939 . . . . . . . 8  |-  F/ x  v R w
132, 12nfrexya 2448 . . . . . . 7  |-  F/ x E. w  e.  A  v R w
1410, 13nfim 1534 . . . . . 6  |-  F/ x
( v R u  ->  E. w  e.  A  v R w )
159, 14nfralya 2447 . . . . 5  |-  F/ x A. v  e.  B  ( v R u  ->  E. w  e.  A  v R w )
168, 15nfan 1527 . . . 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 2585 . . 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 3708 . 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 2252 1  |-  F/_ x sup ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103   F/_wnfc 2242   A.wral 2390   E.wrex 2391   {crab 2394   U.cuni 3702   class class class wbr 3895   supcsup 6821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-sup 6823
This theorem is referenced by:  nfinf  6856  infssuzcldc  11492
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