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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sscon | Structured version Visualization version GIF version |
Description: Contraposition law for relative subclasses. Relative and generalized version of ssconb 4111, which it can shorten, as well as conss2 40652. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssconb 4111 nor conss2 40652. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-sscon | ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4175 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | ineq1i 4182 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ((𝐵 ∩ 𝐴) ∩ 𝑉) |
3 | 2 | eqeq1i 2823 | . 2 ⊢ (((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅ ↔ ((𝐵 ∩ 𝐴) ∩ 𝑉) = ∅) |
4 | bj-disj2r 34237 | . 2 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅) | |
5 | bj-disj2r 34237 | . 2 ⊢ ((𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴) ↔ ((𝐵 ∩ 𝐴) ∩ 𝑉) = ∅) | |
6 | 3, 4, 5 | 3bitr4i 304 | 1 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∖ cdif 3930 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 |
This theorem is referenced by: (None) |
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