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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 4227 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
| 2 | ssdifin0 4486 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → (𝐴 ∩ 𝐴) = ∅) | |
| 3 | 1, 2 | eqtr3id 2791 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → 𝐴 = ∅) | 
| 4 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ (𝐵 ∖ ∅) | |
| 5 | id 22 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
| 6 | difeq2 4120 | . . . 4 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅)) | |
| 7 | 5, 6 | sseq12d 4017 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅))) | 
| 8 | 4, 7 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ (𝐵 ∖ 𝐴)) | 
| 9 | 3, 8 | impbii 209 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 | 
| This theorem is referenced by: disjdifprg 32588 measxun2 34211 measssd 34216 pmeasmono 34326 | 
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