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Theorem nfsup 7093
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 7085 . 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 2347 . . . . . . . 8  |-  F/_ x u
4 nfsup.3 . . . . . . . 8  |-  F/_ x R
5 nfcv 2347 . . . . . . . 8  |-  F/_ x
v
63, 4, 5nfbr 4089 . . . . . . 7  |-  F/ x  u R v
76nfn 1680 . . . . . 6  |-  F/ x  -.  u R v
82, 7nfralya 2545 . . . . 5  |-  F/ x A. v  e.  A  -.  u R v
9 nfsup.2 . . . . . 6  |-  F/_ x B
105, 4, 3nfbr 4089 . . . . . . 7  |-  F/ x  v R u
11 nfcv 2347 . . . . . . . . 9  |-  F/_ x w
125, 4, 11nfbr 4089 . . . . . . . 8  |-  F/ x  v R w
132, 12nfrexya 2546 . . . . . . 7  |-  F/ x E. w  e.  A  v R w
1410, 13nfim 1594 . . . . . 6  |-  F/ x
( v R u  ->  E. w  e.  A  v R w )
159, 14nfralya 2545 . . . . 5  |-  F/ x A. v  e.  B  ( v R u  ->  E. w  e.  A  v R w )
168, 15nfan 1587 . . . 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, 9nfrabw 2686 . . 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 3855 . 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 2344 1  |-  F/_ x sup ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104   F/_wnfc 2334   A.wral 2483   E.wrex 2484   {crab 2487   U.cuni 3849   class class class wbr 4043   supcsup 7083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-sup 7085
This theorem is referenced by:  nfinf  7118  infssuzcldc  10376
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