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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 5041 | . 2 ⊢ (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵)) | |
2 | moeq 3642 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
3 | elsni 4578 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐴) |
5 | 4 | moimi 2545 | . . 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 1802 | 1 ⊢ Disj 𝑥 ∈ {𝐴}𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∃*wmo 2538 {csn 4561 Disj wdisj 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-rmo 3071 df-sn 4562 df-disj 5040 |
This theorem is referenced by: disjx0 5068 disjdifprg 30914 rossros 32148 meadjun 44000 |
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