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Mirrors > Home > NFE Home > Th. List > disj3 | GIF version |
Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.) |
Ref | Expression |
---|---|
disj3 | ⊢ ((A ∩ B) = ∅ ↔ A = (A ∖ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.71 611 | . . . 4 ⊢ ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ A ↔ (x ∈ A ∧ ¬ x ∈ B))) | |
2 | eldif 3221 | . . . . 5 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B)) | |
3 | 2 | bibi2i 304 | . . . 4 ⊢ ((x ∈ A ↔ x ∈ (A ∖ B)) ↔ (x ∈ A ↔ (x ∈ A ∧ ¬ x ∈ B))) |
4 | 1, 3 | bitr4i 243 | . . 3 ⊢ ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ A ↔ x ∈ (A ∖ B))) |
5 | 4 | albii 1566 | . 2 ⊢ (∀x(x ∈ A → ¬ x ∈ B) ↔ ∀x(x ∈ A ↔ x ∈ (A ∖ B))) |
6 | disj1 3593 | . 2 ⊢ ((A ∩ B) = ∅ ↔ ∀x(x ∈ A → ¬ x ∈ B)) | |
7 | dfcleq 2347 | . 2 ⊢ (A = (A ∖ B) ↔ ∀x(x ∈ A ↔ x ∈ (A ∖ B))) | |
8 | 5, 6, 7 | 3bitr4i 268 | 1 ⊢ ((A ∩ B) = ∅ ↔ A = (A ∖ B)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 = wceq 1642 ∈ wcel 1710 ∖ cdif 3206 ∩ cin 3208 ∅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-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 |
This theorem is referenced by: disjel 3597 disj4 3599 uneqdifeq 3638 difprsn1 3847 diftpsn3 3849 ssunsn2 3865 |
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