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Mirrors > Home > NFE Home > Th. List > disj | GIF version |
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
disj | ⊢ ((A ∩ B) = ∅ ↔ ∀x ∈ A ¬ x ∈ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3219 | . . . . 5 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B)) | |
2 | df-an 360 | . . . . 5 ⊢ ((x ∈ A ∧ x ∈ B) ↔ ¬ (x ∈ A → ¬ x ∈ B)) | |
3 | 1, 2 | bitr2i 241 | . . . 4 ⊢ (¬ (x ∈ A → ¬ x ∈ B) ↔ x ∈ (A ∩ B)) |
4 | 3 | con1bii 321 | . . 3 ⊢ (¬ x ∈ (A ∩ B) ↔ (x ∈ A → ¬ x ∈ B)) |
5 | 4 | albii 1566 | . 2 ⊢ (∀x ¬ x ∈ (A ∩ B) ↔ ∀x(x ∈ A → ¬ x ∈ B)) |
6 | eq0 3564 | . 2 ⊢ ((A ∩ B) = ∅ ↔ ∀x ¬ x ∈ (A ∩ B)) | |
7 | df-ral 2619 | . 2 ⊢ (∀x ∈ A ¬ x ∈ B ↔ ∀x(x ∈ A → ¬ x ∈ B)) | |
8 | 5, 6, 7 | 3bitr4i 268 | 1 ⊢ ((A ∩ B) = ∅ ↔ ∀x ∈ A ¬ x ∈ B) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 = wceq 1642 ∈ wcel 1710 ∀wral 2614 ∩ cin 3208 ∅c0 3550 |
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-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: disjr 3592 disj1 3593 disjne 3596 disj5 3890 pw10 4161 pw1disj 4167 phidisjnn 4615 fvun1 5379 xpnedisj 5513 disjex 5823 |
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