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| Mirrors > Home > MPE Home > Th. List > Mathboxes > disjuniel | 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 |
|---|---|
| disjuniel.1 | ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥) |
| disjuniel.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| disjuniel.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵)) |
| Ref | Expression |
|---|---|
| disjuniel | ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uniiun 5058 | . . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥 | |
| 2 | 1 | ineq1i 4216 | . 2 ⊢ (∪ 𝐵 ∩ 𝐶) = (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) |
| 3 | disjuniel.1 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥) | |
| 4 | id 22 | . . 3 ⊢ (𝑥 = 𝐶 → 𝑥 = 𝐶) | |
| 5 | disjuniel.2 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
| 6 | disjuniel.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵)) | |
| 7 | 3, 4, 5, 6 | disjiunel 32609 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) = ∅) |
| 8 | 2, 7 | eqtrid 2789 | 1 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 ∪ 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-uni 4908 df-iun 4993 df-disj 5111 |
| This theorem is referenced by: carsgclctunlem1 34319 |
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