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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 3145 | . 2 ⊢ (𝐴 ∪ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)} | |
2 | nfun.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2323 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfun.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2323 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
6 | 3, 5 | nfor 1584 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵) |
7 | 6 | nfab 2334 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2326 | 1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 709 ∈ wcel 2158 {cab 2173 Ⅎwnfc 2316 ∪ cun 3139 |
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-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-un 3145 |
This theorem is referenced by: nfsuc 4420 nfdju 7055 |
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