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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjun0 | Structured version Visualization version GIF version |
Description: Adding the empty element preserves disjointness. (Contributed by Thierry Arnoux, 30-May-2020.) |
Ref | Expression |
---|---|
disjun0 | ⊢ (Disj 𝑥 ∈ 𝐴 𝑥 → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4733 | . . . . 5 ⊢ (∅ ∈ 𝐴 → {∅} ⊆ 𝐴) | |
2 | ssequn2 4156 | . . . . 5 ⊢ ({∅} ⊆ 𝐴 ↔ (𝐴 ∪ {∅}) = 𝐴) | |
3 | 1, 2 | sylib 219 | . . . 4 ⊢ (∅ ∈ 𝐴 → (𝐴 ∪ {∅}) = 𝐴) |
4 | 3 | disjeq1d 5030 | . . 3 ⊢ (∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ Disj 𝑥 ∈ 𝐴 𝑥)) |
5 | 4 | biimparc 480 | . 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥) |
6 | simpl 483 | . . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ 𝐴 𝑥) | |
7 | in0 4342 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅ | |
8 | 7 | a1i 11 | . . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅) |
9 | 0ex 5202 | . . . . 5 ⊢ ∅ ∈ V | |
10 | id 22 | . . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅) | |
11 | 10 | disjunsn 30272 | . . . . 5 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥 ∈ 𝐴 𝑥 ∧ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅))) |
12 | 9, 11 | mpan 686 | . . . 4 ⊢ (¬ ∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥 ∈ 𝐴 𝑥 ∧ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅))) |
13 | 12 | adantl 482 | . . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥 ∈ 𝐴 𝑥 ∧ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅))) |
14 | 6, 8, 13 | mpbir2and 709 | . 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥) |
15 | 5, 14 | pm2.61dan 809 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝑥 → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 {csn 4557 ∪ ciun 4910 Disj wdisj 5022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-iun 4912 df-disj 5023 |
This theorem is referenced by: carsggect 31475 |
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