Theorem nfsup 6969

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 6961	. 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 2312	. . . . . . . 8 |- F/_ x u
4		nfsup.3	. . . . . . . 8 |- F/_ x R
5		nfcv 2312	. . . . . . . 8 |- F/_ x v
6	3, 4, 5	nfbr 4035	. . . . . . 7 |- F/ x u R v
7	6	nfn 1651	. . . . . 6 |- F/ x -. u R v
8	2, 7	nfralya 2510	. . . . 5 |- F/ x A. v e. A -. u R v
9		nfsup.2	. . . . . 6 |- F/_ x B
10	5, 4, 3	nfbr 4035	. . . . . . 7 |- F/ x v R u
11		nfcv 2312	. . . . . . . . 9 |- F/_ x w
12	5, 4, 11	nfbr 4035	. . . . . . . 8 |- F/ x v R w
13	2, 12	nfrexya 2511	. . . . . . 7 |- F/ x E. w e. A v R w
14	10, 13	nfim 1565	. . . . . 6 |- F/ x ( v R u -> E. w e. A v R w )
15	9, 14	nfralya 2510	. . . . 5 |- F/ x A. v e. B ( v R u -> E. w e. A v R w )
16	8, 15	nfan 1558	. . . 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 2650	. . 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 3802	. 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 2309	1 |- F/_ x sup ( A , B , R )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   F/_wnfc 2299   A.wral 2448   E.wrex 2449   {crab 2452   U.cuni 3796   class class class wbr 3989   supcsup 6959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-sup 6961

This theorem is referenced by:  nfinf  6994  infssuzcldc  11906