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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 4727 | . . . . 5 ⊢ (∅ ∈ 𝐴 → {∅} ⊆ 𝐴) | |
2 | ssequn2 4147 | . . . . 5 ⊢ ({∅} ⊆ 𝐴 ↔ (𝐴 ∪ {∅}) = 𝐴) | |
3 | 1, 2 | sylib 220 | . . . 4 ⊢ (∅ ∈ 𝐴 → (𝐴 ∪ {∅}) = 𝐴) |
4 | 3 | disjeq1d 5025 | . . 3 ⊢ (∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ Disj 𝑥 ∈ 𝐴 𝑥)) |
5 | 4 | biimparc 482 | . 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥) |
6 | simpl 485 | . . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ 𝐴 𝑥) | |
7 | in0 4331 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅ | |
8 | 7 | a1i 11 | . . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅) |
9 | 0ex 5197 | . . . . 5 ⊢ ∅ ∈ V | |
10 | id 22 | . . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅) | |
11 | 10 | disjunsn 30330 | . . . . 5 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥 ∈ 𝐴 𝑥 ∧ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅))) |
12 | 9, 11 | mpan 688 | . . . 4 ⊢ (¬ ∅ ∈ 𝐴 → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥 ∈ 𝐴 𝑥 ∧ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅))) |
13 | 12 | adantl 484 | . . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥 ↔ (Disj 𝑥 ∈ 𝐴 𝑥 ∧ (∪ 𝑥 ∈ 𝐴 𝑥 ∩ ∅) = ∅))) |
14 | 6, 8, 13 | mpbir2and 711 | . 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ ¬ ∅ ∈ 𝐴) → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥) |
15 | 5, 14 | pm2.61dan 811 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝑥 → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3486 ∪ cun 3922 ∩ cin 3923 ⊆ wss 3924 ∅c0 4279 {csn 4553 ∪ ciun 4905 Disj wdisj 5017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5196 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-sn 4554 df-iun 4907 df-disj 5018 |
This theorem is referenced by: carsggect 31583 |
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