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Theorem nfsup 9492
Description: Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
Hypotheses
Ref Expression
nfsup.1 𝑥𝐴
nfsup.2 𝑥𝐵
nfsup.3 𝑥𝑅
Assertion
Ref Expression
nfsup 𝑥sup(𝐴, 𝐵, 𝑅)

Proof of Theorem nfsup
StepHypRef Expression
1 dfsup2 9485 . 2 sup(𝐴, 𝐵, 𝑅) = (𝐵 ∖ ((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴)))))
2 nfsup.2 . . . 4 𝑥𝐵
3 nfsup.3 . . . . . . 7 𝑥𝑅
43nfcnv 5888 . . . . . 6 𝑥𝑅
5 nfsup.1 . . . . . 6 𝑥𝐴
64, 5nfima 6085 . . . . 5 𝑥(𝑅𝐴)
72, 6nfdif 4128 . . . . . 6 𝑥(𝐵 ∖ (𝑅𝐴))
83, 7nfima 6085 . . . . 5 𝑥(𝑅 “ (𝐵 ∖ (𝑅𝐴)))
96, 8nfun 4169 . . . 4 𝑥((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴))))
102, 9nfdif 4128 . . 3 𝑥(𝐵 ∖ ((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴)))))
1110nfuni 4913 . 2 𝑥 (𝐵 ∖ ((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴)))))
121, 11nfcxfr 2902 1 𝑥sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2889  cdif 3947  cun 3948   cuni 4906  ccnv 5683  cima 5687  supcsup 9481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-sup 9483
This theorem is referenced by:  nfinf  9523  itg2cnlem1  25797  esum2d  34095  nfwlim  35824  totbndbnd  37797  aomclem8  43078  binomcxplemdvbinom  44377  binomcxplemdvsum  44379  binomcxplemnotnn0  44380  ssfiunibd  45326  uzub  45447  limsupubuz  45733  fourierdlem20  46147  fourierdlem31  46158  fourierdlem79  46205  sge0ltfirp  46420  pimdecfgtioc  46735  decsmflem  46786  smfsup  46834  smfsupxr  46836  smflimsup  46848
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