![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3359 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | ssdifin0 3519 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → (𝐴 ∩ 𝐴) = ∅) | |
3 | 1, 2 | eqtr3id 2236 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → 𝐴 = ∅) |
4 | 0ss 3476 | . . 3 ⊢ ∅ ⊆ (𝐵 ∖ ∅) | |
5 | id 19 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
6 | difeq2 3262 | . . . 4 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅)) | |
7 | 5, 6 | sseq12d 3201 | . . 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 1364 ∖ cdif 3141 ∩ cin 3143 ⊆ wss 3144 ∅c0 3437 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rab 2477 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-nul 3438 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |