Theorem nfsup 7120

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 7112	. 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))}
2		nfsup.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3		nfcv 2350	. . . . . . . 8 ⊢ Ⅎ𝑥𝑢
4		nfsup.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
5		nfcv 2350	. . . . . . . 8 ⊢ Ⅎ𝑥𝑣
6	3, 4, 5	nfbr 4106	. . . . . . 7 ⊢ Ⅎ𝑥 𝑢𝑅𝑣
7	6	nfn 1682	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑢𝑅𝑣
8	2, 7	nfralya 2548	. . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣
9		nfsup.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
10	5, 4, 3	nfbr 4106	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢
11		nfcv 2350	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑤
12	5, 4, 11	nfbr 4106	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑣𝑅𝑤
13	2, 12	nfrexya 2549	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑤 ∈ 𝐴 𝑣𝑅𝑤
14	10, 13	nfim 1596	. . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤)
15	9, 14	nfralya 2548	. . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤)
16	8, 15	nfan 1589	. . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))
17	16, 9	nfrabw 2689	. . 3 ⊢ Ⅎ𝑥{𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))}
18	17	nfuni 3870	. 2 ⊢ Ⅎ𝑥∪ {𝑢 ∈ 𝐵 ∣ (∀𝑣 ∈ 𝐴 ¬ 𝑢𝑅𝑣 ∧ ∀𝑣 ∈ 𝐵 (𝑣𝑅𝑢 → ∃𝑤 ∈ 𝐴 𝑣𝑅𝑤))}
19	1, 18	nfcxfr 2347	1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  Ⅎwnfc 2337  ∀wral 2486  ∃wrex 2487  {crab 2490  ∪ cuni 3864   class class class wbr 4059  supcsup 7110

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-sup 7112

This theorem is referenced by:  nfinf  7145  infssuzcldc  10415