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Theorem nfsup 7094
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 7086 . 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 2348 . . . . . . . 8  |-  F/_ x u
4 nfsup.3 . . . . . . . 8  |-  F/_ x R
5 nfcv 2348 . . . . . . . 8  |-  F/_ x
v
63, 4, 5nfbr 4090 . . . . . . 7  |-  F/ x  u R v
76nfn 1681 . . . . . 6  |-  F/ x  -.  u R v
82, 7nfralya 2546 . . . . 5  |-  F/ x A. v  e.  A  -.  u R v
9 nfsup.2 . . . . . 6  |-  F/_ x B
105, 4, 3nfbr 4090 . . . . . . 7  |-  F/ x  v R u
11 nfcv 2348 . . . . . . . . 9  |-  F/_ x w
125, 4, 11nfbr 4090 . . . . . . . 8  |-  F/ x  v R w
132, 12nfrexya 2547 . . . . . . 7  |-  F/ x E. w  e.  A  v R w
1410, 13nfim 1595 . . . . . 6  |-  F/ x
( v R u  ->  E. w  e.  A  v R w )
159, 14nfralya 2546 . . . . 5  |-  F/ x A. v  e.  B  ( v R u  ->  E. w  e.  A  v R w )
168, 15nfan 1588 . . . 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 2687 . . 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 3856 . 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 2345 1  |-  F/_ x sup ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104   F/_wnfc 2335   A.wral 2484   E.wrex 2485   {crab 2488   U.cuni 3850   class class class wbr 4044   supcsup 7084
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-sup 7086
This theorem is referenced by:  nfinf  7119  infssuzcldc  10378
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