![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > disjxsn | Structured version Visualization version 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 5119 | . 2 ⊢ (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵)) | |
2 | moeq 3704 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
3 | elsni 4649 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
4 | 3 | adantr 479 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐴) |
5 | 4 | moimi 2534 | . . 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 1793 | 1 ⊢ Disj 𝑥 ∈ {𝐴}𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃*wmo 2527 {csn 4632 Disj wdisj 5117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-mo 2529 df-clab 2706 df-cleq 2720 df-clel 2806 df-rmo 3374 df-sn 4633 df-disj 5118 |
This theorem is referenced by: disjx0 5146 disjdifprg 32386 rossros 33832 meadjun 45879 |
Copyright terms: Public domain | W3C validator |