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Mirrors > Home > ILE Home > Th. List > disjxsn | GIF version |
Description: A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjxsn | ⊢ Disj 𝑥 ∈ {𝐴}𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisj2 3908 | . 2 ⊢ (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵)) | |
2 | moeq 2859 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
3 | elsni 3545 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
4 | 3 | adantr 274 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐴) |
5 | 4 | moimi 2064 | . . 3 ⊢ (∃*𝑥 𝑥 = 𝐴 → ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵)) |
6 | 2, 5 | ax-mp 5 | . 2 ⊢ ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) |
7 | 1, 6 | mpgbir 1429 | 1 ⊢ Disj 𝑥 ∈ {𝐴}𝐵 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∃*wmo 2000 {csn 3527 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-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rmo 2424 df-v 2688 df-sn 3533 df-disj 3907 |
This theorem is referenced by: disjx0 3928 |
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