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Mirrors > Home > MPE Home > Th. List > Mathboxes > inindif | Structured version Visualization version GIF version |
Description: See inundif 4046. (Contributed by Thierry Arnoux, 13-Sep-2017.) |
Ref | Expression |
---|---|
inindif | ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 3834 | . . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶 | |
2 | 1 | orci 405 | . . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶) |
3 | inss 3842 | . . 3 ⊢ (((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶) → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 |
5 | inssdif0 3947 | . 2 ⊢ (((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅) | |
6 | 4, 5 | mpbi 220 | 1 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 383 = wceq 1483 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 |
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 df-nul 3916 |
This theorem is referenced by: resf1o 29505 gsummptres 29784 indsumin 30084 measunl 30279 carsgclctun 30383 probdif 30482 hgt750lemd 30726 |
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