Theorem nfsup 7058

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 7050	. 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))}
2		nfsup.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝑢
4		nfsup.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
5		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝑣
6	3, 4, 5	nfbr 4079	. . . . . . 7 ⊢ Ⅎ𝑥 𝑢𝑅𝑣
7	6	nfn 1672	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣
8	2, 7	nfralya 2537	. . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣
9		nfsup.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
10	5, 4, 3	nfbr 4079	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢
11		nfcv 2339	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑤
12	5, 4, 11	nfbr 4079	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑣𝑅𝑤
13	2, 12	nfrexya 2538	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑤 ∈ 𝐴 𝑣𝑅𝑤
14	10, 13	nfim 1586	. . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤)
15	9, 14	nfralya 2537	. . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤)
16	8, 15	nfan 1579	. . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))
17	16, 9	nfrabw 2678	. . 3 ⊢ Ⅎ𝑥{𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))}
18	17	nfuni 3845	. 2 ⊢ Ⅎ𝑥∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))}
19	1, 18	nfcxfr 2336	1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  Ⅎwnfc 2326  ∀wral 2475  ∃wrex 2476  {crab 2479  ∪ cuni 3839   class class class wbr 4033  supcsup 7048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-sup 7050

This theorem is referenced by:  nfinf  7083  infssuzcldc  10325