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Mirrors > Home > ILE Home > Th. List > nfdisjv | GIF version |
Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.) |
Ref | Expression |
---|---|
nfdisjv.1 | ⊢ Ⅎ𝑦𝐴 |
nfdisjv.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfdisjv | ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisj2 3908 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
2 | nfcv 2281 | . . . . . 6 ⊢ Ⅎ𝑦𝑥 | |
3 | nfdisjv.1 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
4 | 2, 3 | nfel 2290 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
5 | nfdisjv.2 | . . . . . 6 ⊢ Ⅎ𝑦𝐵 | |
6 | 5 | nfcri 2275 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
7 | 4, 6 | nfan 1544 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
8 | 7 | nfmo 2019 | . . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
9 | 8 | nfal 1555 | . 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
10 | 1, 9 | nfxfr 1450 | 1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∀wal 1329 Ⅎwnf 1436 ∈ wcel 1480 ∃*wmo 2000 Ⅎwnfc 2268 Disj wdisj 3906 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rmo 2424 df-disj 3907 |
This theorem is referenced by: (None) |
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