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Theorem necompl 3544
 Description: A class is not equal to its complement. (Contributed by SF, 11-Jan-2015.)
Assertion
Ref Expression
necompl AA

Proof of Theorem necompl
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 pm5.19 349 . . . . 5 ¬ (x A ↔ ¬ x A)
2 vex 2862 . . . . . . 7 x V
32elcompl 3225 . . . . . 6 (x A ↔ ¬ x A)
43bibi2i 304 . . . . 5 ((x Ax A) ↔ (x A ↔ ¬ x A))
51, 4mtbir 290 . . . 4 ¬ (x Ax A)
6 19.8a 1756 . . . 4 (¬ (x Ax A) → x ¬ (x Ax A))
75, 6ax-mp 8 . . 3 x ¬ (x Ax A)
8 dfcleq 2347 . . . . 5 (A = ∼ Ax(x Ax A))
98necon3abii 2546 . . . 4 (A ≠ ∼ A ↔ ¬ x(x Ax A))
10 exnal 1574 . . . 4 (x ¬ (x Ax A) ↔ ¬ x(x Ax A))
119, 10bitr4i 243 . . 3 (A ≠ ∼ Ax ¬ (x Ax A))
127, 11mpbir 200 . 2 A ≠ ∼ A
1312necomi 2598 1 AA
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541   ∈ wcel 1710   ≠ wne 2516   ∼ ccompl 3205 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212 This theorem is referenced by:  nfunv  5138  endisj  6051  ncaddccl  6144  tcdi  6164
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