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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 4801 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
2 | elex 3212 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | disjdifprg 29388 | . . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ 𝐴 ∈ V) → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥) | |
4 | 1, 2, 3 | syl2anc 693 | . 2 ⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥) |
5 | difin 3861 | . . . . 5 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
6 | 5 | preq1i 4271 | . . . 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 4628 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥 ↔ Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)) |
9 | 4, 8 | mpbid 222 | 1 ⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 {cpr 4179 Disj wdisj 4620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 df-disj 4621 |
This theorem is referenced by: measxun2 30273 |
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