New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > dfss4 | GIF version |
Description: Subclass defined in terms of class difference. See comments under dfun2 3490. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
dfss4 | ⊢ (A ⊆ B ↔ (B ∖ (B ∖ A)) = A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqin2 3474 | . 2 ⊢ (A ⊆ B ↔ (B ∩ A) = A) | |
2 | eldif 3221 | . . . . . . 7 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A)) | |
3 | 2 | notbii 287 | . . . . . 6 ⊢ (¬ x ∈ (B ∖ A) ↔ ¬ (x ∈ B ∧ ¬ x ∈ A)) |
4 | 3 | anbi2i 675 | . . . . 5 ⊢ ((x ∈ B ∧ ¬ x ∈ (B ∖ A)) ↔ (x ∈ B ∧ ¬ (x ∈ B ∧ ¬ x ∈ A))) |
5 | elin 3219 | . . . . . 6 ⊢ (x ∈ (B ∩ A) ↔ (x ∈ B ∧ x ∈ A)) | |
6 | abai 770 | . . . . . 6 ⊢ ((x ∈ B ∧ x ∈ A) ↔ (x ∈ B ∧ (x ∈ B → x ∈ A))) | |
7 | iman 413 | . . . . . . 7 ⊢ ((x ∈ B → x ∈ A) ↔ ¬ (x ∈ B ∧ ¬ x ∈ A)) | |
8 | 7 | anbi2i 675 | . . . . . 6 ⊢ ((x ∈ B ∧ (x ∈ B → x ∈ A)) ↔ (x ∈ B ∧ ¬ (x ∈ B ∧ ¬ x ∈ A))) |
9 | 5, 6, 8 | 3bitri 262 | . . . . 5 ⊢ (x ∈ (B ∩ A) ↔ (x ∈ B ∧ ¬ (x ∈ B ∧ ¬ x ∈ A))) |
10 | 4, 9 | bitr4i 243 | . . . 4 ⊢ ((x ∈ B ∧ ¬ x ∈ (B ∖ A)) ↔ x ∈ (B ∩ A)) |
11 | 10 | difeqri 3387 | . . 3 ⊢ (B ∖ (B ∖ A)) = (B ∩ A) |
12 | 11 | eqeq1i 2360 | . 2 ⊢ ((B ∖ (B ∖ A)) = A ↔ (B ∩ A) = A) |
13 | 1, 12 | bitr4i 243 | 1 ⊢ (A ⊆ B ↔ (B ∖ (B ∖ A)) = A) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∖ cdif 3206 ∩ cin 3208 ⊆ wss 3257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 |
This theorem is referenced by: dfin4 3495 |
Copyright terms: Public domain | W3C validator |