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.

Step	Hyp	Ref	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
6	3, 4, 5	nfbr 3939	. . . . . . 7 |- F/ x u R v
7	6	nfn 1619	. . . . . 6 |- F/ x -. u R v
8	2, 7	nfralya 2447	. . . . 5 |- F/ x A. v e. A -. u R v
9		nfsup.2	. . . . . 6 |- F/_ x B
10	5, 4, 3	nfbr 3939	. . . . . . 7 |- F/ x v R u
11		nfcv 2255	. . . . . . . . 9 |- F/_ x w
12	5, 4, 11	nfbr 3939	. . . . . . . 8 |- F/ x v R w
13	2, 12	nfrexya 2448	. . . . . . 7 |- F/ x E. w e. A v R w
14	10, 13	nfim 1534	. . . . . 6 |- F/ x ( v R u -> E. w e. A v R w )
15	9, 14	nfralya 2447	. . . . 5 |- F/ x A. v e. B ( v R u -> E. w e. A v R w )
16	8, 15	nfan 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 ) )
17	16, 9	nfrabxy 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 ) ) }
18	17	nfuni 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 ) ) }
19	1, 18	nfcxfr 2252	1 |- F/_ x sup ( A , B , R )

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