New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > disjne | GIF version |
Description: Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
disjne | ⊢ (((A ∩ B) = ∅ ∧ C ∈ A ∧ D ∈ B) → C ≠ D) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj 3591 | . . 3 ⊢ ((A ∩ B) = ∅ ↔ ∀x ∈ A ¬ x ∈ B) | |
2 | eleq1 2413 | . . . . . 6 ⊢ (x = C → (x ∈ B ↔ C ∈ B)) | |
3 | 2 | notbid 285 | . . . . 5 ⊢ (x = C → (¬ x ∈ B ↔ ¬ C ∈ B)) |
4 | 3 | rspccva 2954 | . . . 4 ⊢ ((∀x ∈ A ¬ x ∈ B ∧ C ∈ A) → ¬ C ∈ B) |
5 | eleq1a 2422 | . . . . 5 ⊢ (D ∈ B → (C = D → C ∈ B)) | |
6 | 5 | necon3bd 2553 | . . . 4 ⊢ (D ∈ B → (¬ C ∈ B → C ≠ D)) |
7 | 4, 6 | syl5com 26 | . . 3 ⊢ ((∀x ∈ A ¬ x ∈ B ∧ C ∈ A) → (D ∈ B → C ≠ D)) |
8 | 1, 7 | sylanb 458 | . 2 ⊢ (((A ∩ B) = ∅ ∧ C ∈ A) → (D ∈ B → C ≠ D)) |
9 | 8 | 3impia 1148 | 1 ⊢ (((A ∩ B) = ∅ ∧ C ∈ A ∧ D ∈ B) → C ≠ D) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 ∧ w3a 934 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∀wral 2614 ∩ 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-3an 936 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: (None) |
Copyright terms: Public domain | W3C validator |