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Theorem dfun2 4275
Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 4276 and dfss4 4274 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation (class difference). (Contributed by NM, 10-Jun-2004.)
Assertion
Ref Expression
dfun2 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Proof of Theorem dfun2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3481 . . . . . . 7 𝑥 ∈ V
2 eldif 3972 . . . . . . 7 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 709 . . . . . 6 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
43anbi1i 624 . . . . 5 ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 eldif 3972 . . . . 5 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵))
6 ioran 985 . . . . 5 (¬ (𝑥𝐴𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 303 . . . 4 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
87con2bii 357 . . 3 ((𝑥𝐴𝑥𝐵) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
9 eldif 3972 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)))
101, 9mpbiran 709 . . 3 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
118, 10bitr4i 278 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)))
1211uneqri 4165 1 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wo 847   = wceq 1536  wcel 2105  Vcvv 3477  cdif 3959  cun 3960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965  df-un 3967
This theorem is referenced by:  dfun3  4281  dfin3  4282
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