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Mirrors > Home > ILE Home > Th. List > nfun | GIF version |
Description: Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfun.1 | ⊢ Ⅎ𝑥𝐴 |
nfun.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfun | ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-un 3003 | . 2 ⊢ (𝐴 ∪ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)} | |
2 | nfun.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2222 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfun.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2222 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
6 | 3, 5 | nfor 1511 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵) |
7 | 6 | nfab 2233 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2225 | 1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 664 ∈ wcel 1438 {cab 2074 Ⅎwnfc 2215 ∪ cun 2997 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-un 3003 |
This theorem is referenced by: nfsuc 4233 nfdju 6725 |
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