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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 3251 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | ssdifin0 3410 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → (𝐴 ∩ 𝐴) = ∅) | |
3 | 1, 2 | syl5eqr 2161 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → 𝐴 = ∅) |
4 | 0ss 3367 | . . 3 ⊢ ∅ ⊆ (𝐵 ∖ ∅) | |
5 | id 19 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
6 | difeq2 3154 | . . . 4 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅)) | |
7 | 5, 6 | sseq12d 3094 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅))) |
8 | 4, 7 | mpbiri 167 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ (𝐵 ∖ 𝐴)) |
9 | 3, 8 | impbii 125 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1314 ∖ cdif 3034 ∩ cin 3036 ⊆ wss 3037 ∅c0 3329 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rab 2399 df-v 2659 df-dif 3039 df-in 3043 df-ss 3050 df-nul 3330 |
This theorem is referenced by: (None) |
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