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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 5325 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
2 | elex 3499 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | disjdifprg 32595 | . . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ 𝐴 ∈ V) → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥) | |
4 | 1, 2, 3 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥) |
5 | difin 4278 | . . . . 5 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
6 | 5 | preq1i 4741 | . . . 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 5123 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥 ↔ Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)) |
9 | 4, 8 | mpbid 232 | 1 ⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ∩ cin 3962 {cpr 4633 Disj wdisj 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-sn 4632 df-pr 4634 df-disj 5116 |
This theorem is referenced by: measxun2 34191 |
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