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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 3878 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
2 | nfcv 2258 | . . . . . 6 ⊢ Ⅎ𝑦𝑥 | |
3 | nfdisjv.1 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
4 | 2, 3 | nfel 2267 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
5 | nfdisjv.2 | . . . . . 6 ⊢ Ⅎ𝑦𝐵 | |
6 | 5 | nfcri 2252 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
7 | 4, 6 | nfan 1529 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
8 | 7 | nfmo 1997 | . . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
9 | 8 | nfal 1540 | . 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
10 | 1, 9 | nfxfr 1435 | 1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∀wal 1314 Ⅎwnf 1421 ∈ wcel 1465 ∃*wmo 1978 Ⅎwnfc 2245 Disj wdisj 3876 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rmo 2401 df-disj 3877 |
This theorem is referenced by: (None) |
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