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Mathbox for Thierry Arnoux |
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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 4842 | . . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥 | |
2 | 1 | ineq1i 4066 | . 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 30106 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) = ∅) |
8 | 2, 7 | syl5eq 2820 | 1 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ∖ cdif 3820 ∩ cin 3822 ⊆ wss 3823 ∅c0 4172 ∪ cuni 4706 ∪ ciun 4786 Disj wdisj 4891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-sn 4436 df-uni 4707 df-iun 4788 df-disj 4892 |
This theorem is referenced by: carsgclctunlem1 31220 |
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