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Theorem nfsup 6398
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 6390 . 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 2194 . . . . . . . 8  |-  F/_ x u
4 nfsup.3 . . . . . . . 8  |-  F/_ x R
5 nfcv 2194 . . . . . . . 8  |-  F/_ x
v
63, 4, 5nfbr 3836 . . . . . . 7  |-  F/ x  u R v
76nfn 1564 . . . . . 6  |-  F/ x  -.  u R v
82, 7nfralya 2379 . . . . 5  |-  F/ x A. v  e.  A  -.  u R v
9 nfsup.2 . . . . . 6  |-  F/_ x B
105, 4, 3nfbr 3836 . . . . . . 7  |-  F/ x  v R u
11 nfcv 2194 . . . . . . . . 9  |-  F/_ x w
125, 4, 11nfbr 3836 . . . . . . . 8  |-  F/ x  v R w
132, 12nfrexya 2380 . . . . . . 7  |-  F/ x E. w  e.  A  v R w
1410, 13nfim 1480 . . . . . 6  |-  F/ x
( v R u  ->  E. w  e.  A  v R w )
159, 14nfralya 2379 . . . . 5  |-  F/ x A. v  e.  B  ( v R u  ->  E. w  e.  A  v R w )
168, 15nfan 1473 . . . 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 2507 . . 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 3614 . 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 2191 1  |-  F/_ x sup ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101   F/_wnfc 2181   A.wral 2323   E.wrex 2324   {crab 2327   U.cuni 3608   class class class wbr 3792   supcsup 6388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-sup 6390
This theorem is referenced by: (None)
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