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Theorem disjne 3250
 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 3245 . . 3 ((AB) = ∅ ↔ x A ¬ x B)
2 eleq1 2082 . . . . . 6 (x = 𝐶 → (x B𝐶 B))
32notbid 579 . . . . 5 (x = 𝐶 → (¬ x B ↔ ¬ 𝐶 B))
43rspccva 2632 . . . 4 ((x A ¬ x B 𝐶 A) → ¬ 𝐶 B)
5 eleq1a 2091 . . . . 5 (𝐷 B → (𝐶 = 𝐷𝐶 B))
65necon3bd 2226 . . . 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 1087 1 (((AB) = ∅ 𝐶 A 𝐷 B) → 𝐶𝐷)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∧ w3a 873   = wceq 1228   ∈ wcel 1374   ≠ wne 2186  ∀wral 2284   ∩ cin 2893  ∅c0 3201 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-v 2537  df-dif 2897  df-in 2901  df-nul 3202 This theorem is referenced by: (None)
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