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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sscon | Structured version Visualization version GIF version |
Description: Contraposition law for relative subsets. Relative and generalized version of ssconb 3743, which it can shorten, as well as conss2 38647. (Contributed by BJ, 11-Nov-2021.) |
Ref | Expression |
---|---|
bj-sscon | ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3805 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | ineq1i 3810 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ((𝐵 ∩ 𝐴) ∩ 𝑉) |
3 | 2 | eqeq1i 2627 | . 2 ⊢ (((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅ ↔ ((𝐵 ∩ 𝐴) ∩ 𝑉) = ∅) |
4 | bj-disj2r 33013 | . 2 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅) | |
5 | bj-disj2r 33013 | . 2 ⊢ ((𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴) ↔ ((𝐵 ∩ 𝐴) ∩ 𝑉) = ∅) | |
6 | 3, 4, 5 | 3bitr4i 292 | 1 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = 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-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 |
This theorem is referenced by: (None) |
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