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Mirrors > Home > MPE Home > Th. List > difin2 | Structured version Visualization version GIF version |
Description: Represent a class difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
difin2 | ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∖ 𝐵) = ((𝐶 ∖ 𝐵) ∩ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3597 | . . . . 5 ⊢ (𝐴 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) | |
2 | 1 | pm4.71d 666 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))) |
3 | 2 | anbi1d 741 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ ¬ 𝑥 ∈ 𝐵))) |
4 | eldif 3584 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
5 | elin 3796 | . . . 4 ⊢ (𝑥 ∈ ((𝐶 ∖ 𝐵) ∩ 𝐴) ↔ (𝑥 ∈ (𝐶 ∖ 𝐵) ∧ 𝑥 ∈ 𝐴)) | |
6 | eldif 3584 | . . . . 5 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) | |
7 | 6 | anbi1i 731 | . . . 4 ⊢ ((𝑥 ∈ (𝐶 ∖ 𝐵) ∧ 𝑥 ∈ 𝐴) ↔ ((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴)) |
8 | ancom 466 | . . . . 5 ⊢ (((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵))) | |
9 | anass 681 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵))) | |
10 | 8, 9 | bitr4i 267 | . . . 4 ⊢ (((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ ¬ 𝑥 ∈ 𝐵)) |
11 | 5, 7, 10 | 3bitri 286 | . . 3 ⊢ (𝑥 ∈ ((𝐶 ∖ 𝐵) ∩ 𝐴) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ ¬ 𝑥 ∈ 𝐵)) |
12 | 3, 4, 11 | 3bitr4g 303 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 𝑥 ∈ ((𝐶 ∖ 𝐵) ∩ 𝐴))) |
13 | 12 | eqrdv 2620 | 1 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∖ 𝐵) = ((𝐶 ∖ 𝐵) ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 |
This theorem is referenced by: gsumdifsnd 18360 issubdrg 18805 restcld 20976 limcnlp 23642 difelsiga 30196 sigapildsyslem 30224 ldgenpisyslem1 30226 difelcarsg2 30375 ballotlemfp1 30553 asindmre 33495 caragendifcl 40728 gsumdifsndf 42144 |
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