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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 3916 | . 2 ⊢ (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵)) | |
2 | moeq 2863 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
3 | elsni 3550 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
4 | 3 | adantr 274 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐴) |
5 | 4 | moimi 2065 | . . 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 1430 | 1 ⊢ Disj 𝑥 ∈ {𝐴}𝐵 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∈ wcel 1481 ∃*wmo 2001 {csn 3532 Disj wdisj 3914 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rmo 2425 df-v 2691 df-sn 3538 df-disj 3915 |
This theorem is referenced by: disjx0 3936 |
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