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Theorem nfsup 6847
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 6839 . 2 sup(𝐴, 𝐵, 𝑅) = {𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
2 nfsup.1 . . . . . 6 𝑥𝐴
3 nfcv 2258 . . . . . . . 8 𝑥𝑢
4 nfsup.3 . . . . . . . 8 𝑥𝑅
5 nfcv 2258 . . . . . . . 8 𝑥𝑣
63, 4, 5nfbr 3944 . . . . . . 7 𝑥 𝑢𝑅𝑣
76nfn 1621 . . . . . 6 𝑥 ¬ 𝑢𝑅𝑣
82, 7nfralya 2450 . . . . 5 𝑥𝑣𝐴 ¬ 𝑢𝑅𝑣
9 nfsup.2 . . . . . 6 𝑥𝐵
105, 4, 3nfbr 3944 . . . . . . 7 𝑥 𝑣𝑅𝑢
11 nfcv 2258 . . . . . . . . 9 𝑥𝑤
125, 4, 11nfbr 3944 . . . . . . . 8 𝑥 𝑣𝑅𝑤
132, 12nfrexya 2451 . . . . . . 7 𝑥𝑤𝐴 𝑣𝑅𝑤
1410, 13nfim 1536 . . . . . 6 𝑥(𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤)
159, 14nfralya 2450 . . . . 5 𝑥𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤)
168, 15nfan 1529 . . . 4 𝑥(∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))
1716, 9nfrabxy 2588 . . 3 𝑥{𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
1817nfuni 3712 . 2 𝑥 {𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
191, 18nfcxfr 2255 1 𝑥sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wnfc 2245  wral 2393  wrex 2394  {crab 2397   cuni 3706   class class class wbr 3899  supcsup 6837
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-sup 6839
This theorem is referenced by:  nfinf  6872  infssuzcldc  11571
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