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Theorem nfsup 6398
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 6390 . 2 sup(𝐴, 𝐵, 𝑅) = {𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
2 nfsup.1 . . . . . 6 𝑥𝐴
3 nfcv 2194 . . . . . . . 8 𝑥𝑢
4 nfsup.3 . . . . . . . 8 𝑥𝑅
5 nfcv 2194 . . . . . . . 8 𝑥𝑣
63, 4, 5nfbr 3836 . . . . . . 7 𝑥 𝑢𝑅𝑣
76nfn 1564 . . . . . 6 𝑥 ¬ 𝑢𝑅𝑣
82, 7nfralya 2379 . . . . 5 𝑥𝑣𝐴 ¬ 𝑢𝑅𝑣
9 nfsup.2 . . . . . 6 𝑥𝐵
105, 4, 3nfbr 3836 . . . . . . 7 𝑥 𝑣𝑅𝑢
11 nfcv 2194 . . . . . . . . 9 𝑥𝑤
125, 4, 11nfbr 3836 . . . . . . . 8 𝑥 𝑣𝑅𝑤
132, 12nfrexya 2380 . . . . . . 7 𝑥𝑤𝐴 𝑣𝑅𝑤
1410, 13nfim 1480 . . . . . 6 𝑥(𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤)
159, 14nfralya 2379 . . . . 5 𝑥𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤)
168, 15nfan 1473 . . . 4 𝑥(∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))
1716, 9nfrabxy 2507 . . 3 𝑥{𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
1817nfuni 3614 . 2 𝑥 {𝑢𝐵 ∣ (∀𝑣𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣𝐵 (𝑣𝑅𝑢 → ∃𝑤𝐴 𝑣𝑅𝑤))}
191, 18nfcxfr 2191 1 𝑥sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wnfc 2181  wral 2323  wrex 2324  {crab 2327   cuni 3608   class class class wbr 3792  supcsup 6388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-sup 6390
This theorem is referenced by: (None)
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