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| Mirrors > Home > ILE Home > Th. List > ssdifeq0 | 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 3381 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
| 2 | ssdifin0 3541 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → (𝐴 ∩ 𝐴) = ∅) | |
| 3 | 1, 2 | eqtr3id 2251 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → 𝐴 = ∅) |
| 4 | 0ss 3498 | . . 3 ⊢ ∅ ⊆ (𝐵 ∖ ∅) | |
| 5 | id 19 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
| 6 | difeq2 3284 | . . . 4 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅)) | |
| 7 | 5, 6 | sseq12d 3223 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅))) |
| 8 | 4, 7 | mpbiri 168 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ (𝐵 ∖ 𝐴)) |
| 9 | 3, 8 | impbii 126 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) |
| Colors of variables: wff set class |
| Syntax hints: ↔ wb 105 = wceq 1372 ∖ cdif 3162 ∩ cin 3164 ⊆ wss 3165 ∅c0 3459 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rab 2492 df-v 2773 df-dif 3167 df-in 3171 df-ss 3178 df-nul 3460 |
| This theorem is referenced by: (None) |
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