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Theorem reldisj 3265
Description: Two ways of saying that two classes are disjoint, using the complement of B relative to a universe 𝐶. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
reldisj (A𝐶 → ((AB) = ∅ ↔ A ⊆ (𝐶B)))

Proof of Theorem reldisj
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 dfss2 2928 . . . 4 (A𝐶x(x Ax 𝐶))
2 pm5.44 833 . . . . . 6 ((x Ax 𝐶) → ((x A → ¬ x B) ↔ (x A → (x 𝐶 ¬ x B))))
3 eldif 2921 . . . . . . 7 (x (𝐶B) ↔ (x 𝐶 ¬ x B))
43imbi2i 215 . . . . . 6 ((x Ax (𝐶B)) ↔ (x A → (x 𝐶 ¬ x B)))
52, 4syl6bbr 187 . . . . 5 ((x Ax 𝐶) → ((x A → ¬ x B) ↔ (x Ax (𝐶B))))
65sps 1427 . . . 4 (x(x Ax 𝐶) → ((x A → ¬ x B) ↔ (x Ax (𝐶B))))
71, 6sylbi 114 . . 3 (A𝐶 → ((x A → ¬ x B) ↔ (x Ax (𝐶B))))
87albidv 1702 . 2 (A𝐶 → (x(x A → ¬ x B) ↔ x(x Ax (𝐶B))))
9 disj1 3264 . 2 ((AB) = ∅ ↔ x(x A → ¬ x B))
10 dfss2 2928 . 2 (A ⊆ (𝐶B) ↔ x(x Ax (𝐶B)))
118, 9, 103bitr4g 212 1 (A𝐶 → ((AB) = ∅ ↔ A ⊆ (𝐶B)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  cdif 2908  cin 2910  wss 2911  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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219
This theorem is referenced by:  disj2  3269
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