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Theorem nfsup 6886
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 6878 . 2 sup(𝐴, 𝐵, 𝑅) = {𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
2 nfsup.1 . . . . . 6 𝑥𝐴
3 nfcv 2282 . . . . . . . 8 𝑥𝑢
4 nfsup.3 . . . . . . . 8 𝑥𝑅
5 nfcv 2282 . . . . . . . 8 𝑥𝑣
63, 4, 5nfbr 3981 . . . . . . 7 𝑥 𝑢𝑅𝑣
76nfn 1637 . . . . . 6 𝑥 ¬ 𝑢𝑅𝑣
82, 7nfralya 2476 . . . . 5 𝑥𝑣𝐴 ¬ 𝑢𝑅𝑣
9 nfsup.2 . . . . . 6 𝑥𝐵
105, 4, 3nfbr 3981 . . . . . . 7 𝑥 𝑣𝑅𝑢
11 nfcv 2282 . . . . . . . . 9 𝑥𝑤
125, 4, 11nfbr 3981 . . . . . . . 8 𝑥 𝑣𝑅𝑤
132, 12nfrexya 2477 . . . . . . 7 𝑥𝑤𝐴 𝑣𝑅𝑤
1410, 13nfim 1552 . . . . . 6 𝑥(𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤)
159, 14nfralya 2476 . . . . 5 𝑥𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤)
168, 15nfan 1545 . . . 4 𝑥(∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))
1716, 9nfrabxy 2614 . . 3 𝑥{𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
1817nfuni 3749 . 2 𝑥 {𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
191, 18nfcxfr 2279 1 𝑥sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wnfc 2269  wral 2417  wrex 2418  {crab 2421   cuni 3743   class class class wbr 3936  supcsup 6876
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-sup 6878
This theorem is referenced by:  nfinf  6911  infssuzcldc  11678
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