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Theorem nfsup 6969
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 6961 . 2 sup(𝐴, 𝐵, 𝑅) = {𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
2 nfsup.1 . . . . . 6 𝑥𝐴
3 nfcv 2312 . . . . . . . 8 𝑥𝑢
4 nfsup.3 . . . . . . . 8 𝑥𝑅
5 nfcv 2312 . . . . . . . 8 𝑥𝑣
63, 4, 5nfbr 4035 . . . . . . 7 𝑥 𝑢𝑅𝑣
76nfn 1651 . . . . . 6 𝑥 ¬ 𝑢𝑅𝑣
82, 7nfralya 2510 . . . . 5 𝑥𝑣𝐴 ¬ 𝑢𝑅𝑣
9 nfsup.2 . . . . . 6 𝑥𝐵
105, 4, 3nfbr 4035 . . . . . . 7 𝑥 𝑣𝑅𝑢
11 nfcv 2312 . . . . . . . . 9 𝑥𝑤
125, 4, 11nfbr 4035 . . . . . . . 8 𝑥 𝑣𝑅𝑤
132, 12nfrexya 2511 . . . . . . 7 𝑥𝑤𝐴 𝑣𝑅𝑤
1410, 13nfim 1565 . . . . . 6 𝑥(𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤)
159, 14nfralya 2510 . . . . 5 𝑥𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤)
168, 15nfan 1558 . . . 4 𝑥(∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))
1716, 9nfrabxy 2650 . . 3 𝑥{𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
1817nfuni 3802 . 2 𝑥 {𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
191, 18nfcxfr 2309 1 𝑥sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wnfc 2299  wral 2448  wrex 2449  {crab 2452   cuni 3796   class class class wbr 3989  supcsup 6959
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-sup 6961
This theorem is referenced by:  nfinf  6994  infssuzcldc  11906
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