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Mirrors > Home > MPE Home > Th. List > ssdifeq0 | Structured version Visualization version GIF version |
Description: A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.) |
Ref | Expression |
---|---|
ssdifeq0 | ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inidm 4158 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | ssdifin0 4422 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → (𝐴 ∩ 𝐴) = ∅) | |
3 | 1, 2 | eqtr3id 2790 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → 𝐴 = ∅) |
4 | 0ss 4336 | . . 3 ⊢ ∅ ⊆ (𝐵 ∖ ∅) | |
5 | id 22 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
6 | difeq2 4057 | . . . 4 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅)) | |
7 | 5, 6 | sseq12d 3959 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅))) |
8 | 4, 7 | mpbiri 259 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ (𝐵 ∖ 𝐴)) |
9 | 3, 8 | impbii 208 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 ∖ cdif 3889 ∩ cin 3891 ⊆ wss 3892 ∅c0 4262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3287 df-v 3439 df-dif 3895 df-in 3899 df-ss 3909 df-nul 4263 |
This theorem is referenced by: disjdifprg 30955 measxun2 32219 measssd 32224 pmeasmono 32332 |
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