|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > disjiunel | Structured version Visualization version GIF version | ||
| Description: A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.) | 
| Ref | Expression | 
|---|---|
| disjiunel.1 | ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) | 
| disjiunel.2 | ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | 
| disjiunel.3 | ⊢ (𝜑 → 𝐸 ⊆ 𝐴) | 
| disjiunel.4 | ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 𝐸)) | 
| Ref | Expression | 
|---|---|
| disjiunel | ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | disjiunel.3 | . . . . 5 ⊢ (𝜑 → 𝐸 ⊆ 𝐴) | |
| 2 | disjiunel.4 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 𝐸)) | |
| 3 | 2 | eldifad 3963 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | 
| 4 | 3 | snssd 4809 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝐴) | 
| 5 | 1, 4 | unssd 4192 | . . . 4 ⊢ (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴) | 
| 6 | disjiunel.1 | . . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) | |
| 7 | disjss1 5116 | . . . 4 ⊢ ((𝐸 ∪ {𝑌}) ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)) | |
| 8 | 5, 6, 7 | sylc 65 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵) | 
| 9 | 2 | eldifbd 3964 | . . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐸) | 
| 10 | disjiunel.2 | . . . . 5 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | |
| 11 | 10 | disjunsn 32607 | . . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))) | 
| 12 | 3, 9, 11 | syl2anc 584 | . . 3 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))) | 
| 13 | 8, 12 | mpbid 232 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)) | 
| 14 | 13 | simprd 495 | 1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 ∪ ciun 4991 Disj wdisj 5110 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-iun 4993 df-disj 5111 | 
| This theorem is referenced by: disjuniel 32610 | 
| Copyright terms: Public domain | W3C validator |