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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 3747 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
2 | nfcv 2178 | . . . . . 6 ⊢ Ⅎ𝑦𝑥 | |
3 | nfdisjv.1 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
4 | 2, 3 | nfel 2186 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
5 | nfdisjv.2 | . . . . . 6 ⊢ Ⅎ𝑦𝐵 | |
6 | 5 | nfcri 2172 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
7 | 4, 6 | nfan 1457 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
8 | 7 | nfmo 1920 | . . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
9 | 8 | nfal 1468 | . 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
10 | 1, 9 | nfxfr 1363 | 1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∀wal 1241 Ⅎwnf 1349 ∈ wcel 1393 ∃*wmo 1901 Ⅎwnfc 2165 Disj wdisj 3745 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rmo 2314 df-disj 3746 |
This theorem is referenced by: (None) |
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