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Theorem nfsup 7009
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
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sup 7001 . 2 sup(𝐴, 𝐵, 𝑅) = {𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
2 nfsup.1 . . . . . 6 𝑥𝐴
3 nfcv 2332 . . . . . . . 8 𝑥𝑢
4 nfsup.3 . . . . . . . 8 𝑥𝑅
5 nfcv 2332 . . . . . . . 8 𝑥𝑣
63, 4, 5nfbr 4064 . . . . . . 7 𝑥 𝑢𝑅𝑣
76nfn 1669 . . . . . 6 𝑥 ¬ 𝑢𝑅𝑣
82, 7nfralya 2530 . . . . 5 𝑥𝑣𝐴 ¬ 𝑢𝑅𝑣
9 nfsup.2 . . . . . 6 𝑥𝐵
105, 4, 3nfbr 4064 . . . . . . 7 𝑥 𝑣𝑅𝑢
11 nfcv 2332 . . . . . . . . 9 𝑥𝑤
125, 4, 11nfbr 4064 . . . . . . . 8 𝑥 𝑣𝑅𝑤
132, 12nfrexya 2531 . . . . . . 7 𝑥𝑤𝐴 𝑣𝑅𝑤
1410, 13nfim 1583 . . . . . 6 𝑥(𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤)
159, 14nfralya 2530 . . . . 5 𝑥𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤)
168, 15nfan 1576 . . . 4 𝑥(∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))
1716, 9nfrabxy 2671 . . 3 𝑥{𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
1817nfuni 3830 . 2 𝑥 {𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
191, 18nfcxfr 2329 1 𝑥sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wnfc 2319  wral 2468  wrex 2469  {crab 2472   cuni 3824   class class class wbr 4018  supcsup 6999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-sup 7001
This theorem is referenced by:  nfinf  7034  infssuzcldc  11970
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