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Theorem nfsup 8907
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 8900 . 2 sup(𝐴, 𝐵, 𝑅) = (𝐵 ∖ ((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴)))))
2 nfsup.2 . . . 4 𝑥𝐵
3 nfsup.3 . . . . . . 7 𝑥𝑅
43nfcnv 5742 . . . . . 6 𝑥𝑅
5 nfsup.1 . . . . . 6 𝑥𝐴
64, 5nfima 5930 . . . . 5 𝑥(𝑅𝐴)
72, 6nfdif 4100 . . . . . 6 𝑥(𝐵 ∖ (𝑅𝐴))
83, 7nfima 5930 . . . . 5 𝑥(𝑅 “ (𝐵 ∖ (𝑅𝐴)))
96, 8nfun 4139 . . . 4 𝑥((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴))))
102, 9nfdif 4100 . . 3 𝑥(𝐵 ∖ ((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴)))))
1110nfuni 4837 . 2 𝑥 (𝐵 ∖ ((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴)))))
121, 11nfcxfr 2973 1 𝑥sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2959  cdif 3931  cun 3932   cuni 4830  ccnv 5547  cima 5551  supcsup 8896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-sup 8898
This theorem is referenced by:  nfinf  8938  itg2cnlem1  24354  esum2d  31345  nfwlim  33102  totbndbnd  35059  aomclem8  39652  binomcxplemdvbinom  40676  binomcxplemdvsum  40678  binomcxplemnotnn0  40679  ssfiunibd  41566  uzub  41695  limsupubuz  41984  fourierdlem20  42403  fourierdlem31  42414  fourierdlem79  42461  sge0ltfirp  42673  pimdecfgtioc  42984  decsmflem  43033  smfsup  43079  smfsupxr  43081  smflimsup  43093
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