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Mirrors > Home > MPE Home > Th. List > ssdifsym | Structured version Visualization version GIF version |
Description: Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.) |
Ref | Expression |
---|---|
ssdifsym | ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) ↔ 𝐴 = (𝑉 ∖ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdifim 4220 | . . . 4 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵)) | |
2 | 1 | ex 413 | . . 3 ⊢ (𝐴 ⊆ 𝑉 → (𝐵 = (𝑉 ∖ 𝐴) → 𝐴 = (𝑉 ∖ 𝐵))) |
3 | 2 | adantr 481 | . 2 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) → 𝐴 = (𝑉 ∖ 𝐵))) |
4 | ssdifim 4220 | . . . 4 ⊢ ((𝐵 ⊆ 𝑉 ∧ 𝐴 = (𝑉 ∖ 𝐵)) → 𝐵 = (𝑉 ∖ 𝐴)) | |
5 | 4 | ex 413 | . . 3 ⊢ (𝐵 ⊆ 𝑉 → (𝐴 = (𝑉 ∖ 𝐵) → 𝐵 = (𝑉 ∖ 𝐴))) |
6 | 5 | adantl 482 | . 2 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐴 = (𝑉 ∖ 𝐵) → 𝐵 = (𝑉 ∖ 𝐴))) |
7 | 3, 6 | impbid 211 | 1 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) ↔ 𝐴 = (𝑉 ∖ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∖ cdif 3905 ⊆ wss 3908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-in 3915 df-ss 3925 |
This theorem is referenced by: zarcls 32324 |
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