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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjdifprg2 | Structured version Visualization version GIF version |
Description: A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
disjdifprg2 | ⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inex1g 5318 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
2 | elex 3491 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | disjdifprg 32073 | . . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ 𝐴 ∈ V) → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥) | |
4 | 1, 2, 3 | syl2anc 582 | . 2 ⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥) |
5 | difin 4260 | . . . . 5 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
6 | 5 | preq1i 4739 | . . . 4 ⊢ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)} |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}) |
8 | 7 | disjeq1d 5120 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥 ↔ Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)) |
9 | 4, 8 | mpbid 231 | 1 ⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∖ cdif 3944 ∩ cin 3946 {cpr 4629 Disj wdisj 5112 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-sn 4628 df-pr 4630 df-disj 5113 |
This theorem is referenced by: measxun2 33506 |
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