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Theorem dfun2 3892
Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3893 and dfss4 3891 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 3234 . . . . . . 7 𝑥 ∈ V
2 eldif 3617 . . . . . . 7 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 973 . . . . . 6 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
43anbi1i 731 . . . . 5 ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 eldif 3617 . . . . 5 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵))
6 ioran 510 . . . . 5 (¬ (𝑥𝐴𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 292 . . . 4 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
87con2bii 346 . . 3 ((𝑥𝐴𝑥𝐵) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
9 eldif 3617 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)))
101, 9mpbiran 973 . . 3 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
118, 10bitr4i 267 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)))
1211uneqri 3788 1 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  cun 3605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612
This theorem is referenced by:  dfun3  3898  dfin3  3899
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