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Mirrors > Home > ILE Home > Th. List > nfdju | GIF version |
Description: Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
Ref | Expression |
---|---|
nfdju.1 | ⊢ Ⅎ𝑥𝐴 |
nfdju.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfdju | ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 6916 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | nfcv 2279 | . . . 4 ⊢ Ⅎ𝑥{∅} | |
3 | nfdju.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfxp 4561 | . . 3 ⊢ Ⅎ𝑥({∅} × 𝐴) |
5 | nfcv 2279 | . . . 4 ⊢ Ⅎ𝑥{1o} | |
6 | nfdju.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfxp 4561 | . . 3 ⊢ Ⅎ𝑥({1o} × 𝐵) |
8 | 4, 7 | nfun 3227 | . 2 ⊢ Ⅎ𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵)) |
9 | 1, 8 | nfcxfr 2276 | 1 ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵) |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2266 ∪ cun 3064 ∅c0 3358 {csn 3522 × cxp 4532 1oc1o 6299 ⊔ cdju 6915 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-un 3070 df-opab 3985 df-xp 4540 df-dju 6916 |
This theorem is referenced by: (None) |
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