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Theorem disj 3379
 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) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disj
StepHypRef Expression
1 df-in 3045 . . . 4 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
21eqeq1i 2123 . . 3 ((𝐴𝐵) = ∅ ↔ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = ∅)
3 abeq1 2225 . . 3 ({𝑥 ∣ (𝑥𝐴𝑥𝐵)} = ∅ ↔ ∀𝑥((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅))
4 imnan 662 . . . . 5 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
5 noel 3335 . . . . . 6 ¬ 𝑥 ∈ ∅
65nbn 671 . . . . 5 (¬ (𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅))
74, 6bitr2i 184 . . . 4 (((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥𝐴 → ¬ 𝑥𝐵))
87albii 1429 . . 3 (∀𝑥((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
92, 3, 83bitri 205 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
10 df-ral 2396 . 2 (∀𝑥𝐴 ¬ 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
119, 10bitr4i 186 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1312   = wceq 1314   ∈ wcel 1463  {cab 2101  ∀wral 2391   ∩ cin 3038  ∅c0 3331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-dif 3041  df-in 3045  df-nul 3332 This theorem is referenced by:  disjr  3380  disj1  3381  disjne  3384  f0rn0  5285  renfdisj  7788  fvinim0ffz  9958  fxnn0nninf  10151  exmidunben  11834  dedekindeulemuub  12659  dedekindeulemlu  12663  dedekindicclemuub  12668  dedekindicclemlu  12672  ivthinclemdisj  12682  exmidsbthrlem  13040
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