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| 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 5111 | . 2 ⊢ (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵)) | |
| 2 | moeq 3712 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
| 3 | elsni 4642 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
| 4 | 3 | adantr 480 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐴) | 
| 5 | 4 | moimi 2544 | . . 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 1798 | 1 ⊢ Disj 𝑥 ∈ {𝐴}𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃*wmo 2537 {csn 4625 Disj wdisj 5109 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-rmo 3379 df-sn 4626 df-disj 5110 | 
| This theorem is referenced by: disjx0 5137 disjdifprg 32589 rossros 34182 meadjun 46482 | 
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