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Theorem disj 3262
 Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
Assertion
Ref Expression
disj ((AB) = ∅ ↔ x A ¬ x B)
Distinct variable groups:   x,A   x,B

Proof of Theorem disj
StepHypRef Expression
1 df-in 2918 . . . 4 (AB) = {x ∣ (x A x B)}
21eqeq1i 2044 . . 3 ((AB) = ∅ ↔ {x ∣ (x A x B)} = ∅)
3 abeq1 2144 . . 3 ({x ∣ (x A x B)} = ∅ ↔ x((x A x B) ↔ x ∅))
4 imnan 623 . . . . 5 ((x A → ¬ x B) ↔ ¬ (x A x B))
5 noel 3222 . . . . . 6 ¬ x
65nbn 614 . . . . 5 (¬ (x A x B) ↔ ((x A x B) ↔ x ∅))
74, 6bitr2i 174 . . . 4 (((x A x B) ↔ x ∅) ↔ (x A → ¬ x B))
87albii 1356 . . 3 (x((x A x B) ↔ x ∅) ↔ x(x A → ¬ x B))
92, 3, 83bitri 195 . 2 ((AB) = ∅ ↔ x(x A → ¬ x B))
10 df-ral 2305 . 2 (x A ¬ x Bx(x A → ¬ x B))
119, 10bitr4i 176 1 ((AB) = ∅ ↔ x A ¬ x B)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀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-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-nul 3219 This theorem is referenced by:  disjr  3263  disj1  3264  disjne  3267  renfdisj  6856
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