Theorem nfsup 6948

Description: Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)

Hypotheses

Ref	Expression
nfsup.1	⊢ Ⅎ𝑥𝐴
nfsup.2	⊢ Ⅎ𝑥𝐵
nfsup.3	⊢ Ⅎ𝑥𝑅

Assertion

Ref	Expression
nfsup	⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅)

Proof of Theorem nfsup

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sup 6940	. 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))}
2		nfsup.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3		nfcv 2306	. . . . . . . 8 ⊢ Ⅎ𝑥𝑢
4		nfsup.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
5		nfcv 2306	. . . . . . . 8 ⊢ Ⅎ𝑥𝑣
6	3, 4, 5	nfbr 4022	. . . . . . 7 ⊢ Ⅎ𝑥 𝑢𝑅𝑣
7	6	nfn 1645	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣
8	2, 7	nfralya 2504	. . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣
9		nfsup.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
10	5, 4, 3	nfbr 4022	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢
11		nfcv 2306	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑤
12	5, 4, 11	nfbr 4022	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑣𝑅𝑤
13	2, 12	nfrexya 2505	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑤 ∈ 𝐴 𝑣𝑅𝑤
14	10, 13	nfim 1559	. . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤)
15	9, 14	nfralya 2504	. . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤)
16	8, 15	nfan 1552	. . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))
17	16, 9	nfrabxy 2644	. . 3 ⊢ Ⅎ𝑥{𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))}
18	17	nfuni 3789	. 2 ⊢ Ⅎ𝑥∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))}
19	1, 18	nfcxfr 2303	1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  Ⅎwnfc 2293  ∀wral 2442  ∃wrex 2443  {crab 2446  ∪ cuni 3783   class class class wbr 3976  supcsup 6938

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-sup 6940

This theorem is referenced by:  nfinf  6973  infssuzcldc  11869