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Mirrors > Home > NFE 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 3464 | . . 3 ⊢ (A ∩ A) = A | |
2 | ssdifin0 3631 | . . 3 ⊢ (A ⊆ (B ∖ A) → (A ∩ A) = ∅) | |
3 | 1, 2 | syl5eqr 2399 | . 2 ⊢ (A ⊆ (B ∖ A) → A = ∅) |
4 | 0ss 3579 | . . 3 ⊢ ∅ ⊆ (B ∖ ∅) | |
5 | id 19 | . . . 4 ⊢ (A = ∅ → A = ∅) | |
6 | difeq2 3247 | . . . 4 ⊢ (A = ∅ → (B ∖ A) = (B ∖ ∅)) | |
7 | 5, 6 | sseq12d 3300 | . . 3 ⊢ (A = ∅ → (A ⊆ (B ∖ A) ↔ ∅ ⊆ (B ∖ ∅))) |
8 | 4, 7 | mpbiri 224 | . 2 ⊢ (A = ∅ → A ⊆ (B ∖ A)) |
9 | 3, 8 | impbii 180 | 1 ⊢ (A ⊆ (B ∖ A) ↔ A = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 ∖ cdif 3206 ∩ cin 3208 ⊆ wss 3257 ∅c0 3550 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: (None) |
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