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Theorem reldisj 3248
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 2911 . . . 4 (A𝐶x(x Ax 𝐶))
2 pm5.44 822 . . . . . 6 ((x Ax 𝐶) → ((x A → ¬ x B) ↔ (x A → (x 𝐶 ¬ x B))))
3 eldif 2904 . . . . . . 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 1412 . . . 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 1687 . 2 (A𝐶 → (x(x A → ¬ x B) ↔ x(x Ax (𝐶B))))
9 disj1 3247 . 2 ((AB) = ∅ ↔ x(x A → ¬ x B))
10 dfss2 2911 . 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 1226   = wceq 1228   wcel 1374  cdif 2891  cin 2893  wss 2894  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-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908  df-nul 3202
This theorem is referenced by:  disj2  3252
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