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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 5117 | . 2 ⊢ (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵)) | |
2 | moeq 3716 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
3 | elsni 4648 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐴) |
5 | 4 | moimi 2543 | . . 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 1796 | 1 ⊢ Disj 𝑥 ∈ {𝐴}𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃*wmo 2536 {csn 4631 Disj wdisj 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-rmo 3378 df-sn 4632 df-disj 5116 |
This theorem is referenced by: disjx0 5143 disjdifprg 32595 rossros 34161 meadjun 46418 |
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