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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 4114, which it can shorten, as well as conss2 40795. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssconb 4114 nor conss2 40795. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-sscon | ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4178 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | ineq1i 4185 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ((𝐵 ∩ 𝐴) ∩ 𝑉) |
3 | 2 | eqeq1i 2826 | . 2 ⊢ (((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅ ↔ ((𝐵 ∩ 𝐴) ∩ 𝑉) = ∅) |
4 | bj-disj2r 34343 | . 2 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅) | |
5 | bj-disj2r 34343 | . 2 ⊢ ((𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴) ↔ ((𝐵 ∩ 𝐴) ∩ 𝑉) = ∅) | |
6 | 3, 4, 5 | 3bitr4i 305 | 1 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-in 3943 df-ss 3952 df-nul 4292 |
This theorem is referenced by: (None) |
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