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Theorem disjne 3267
Description: Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
disjne (((AB) = ∅ 𝐶 A 𝐷 B) → 𝐶𝐷)

Proof of Theorem disjne
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 disj 3262 . . 3 ((AB) = ∅ ↔ x A ¬ x B)
2 eleq1 2097 . . . . . 6 (x = 𝐶 → (x B𝐶 B))
32notbid 591 . . . . 5 (x = 𝐶 → (¬ x B ↔ ¬ 𝐶 B))
43rspccva 2649 . . . 4 ((x A ¬ x B 𝐶 A) → ¬ 𝐶 B)
5 eleq1a 2106 . . . . 5 (𝐷 B → (𝐶 = 𝐷𝐶 B))
65necon3bd 2242 . . . 4 (𝐷 B → (¬ 𝐶 B𝐶𝐷))
74, 6syl5com 26 . . 3 ((x A ¬ x B 𝐶 A) → (𝐷 B𝐶𝐷))
81, 7sylanb 268 . 2 (((AB) = ∅ 𝐶 A) → (𝐷 B𝐶𝐷))
983impia 1100 1 (((AB) = ∅ 𝐶 A 𝐷 B) → 𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wne 2201  wral 2300  cin 2910  c0 3218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-nul 3219
This theorem is referenced by: (None)
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