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Theorem dfnul3 3425
Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
dfnul3 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}

Proof of Theorem dfnul3
StepHypRef Expression
1 equid 1701 . . . . 5 𝑥 = 𝑥
21notnoti 645 . . . 4 ¬ ¬ 𝑥 = 𝑥
3 pm3.24 693 . . . 4 ¬ (𝑥𝐴 ∧ ¬ 𝑥𝐴)
42, 32false 701 . . 3 𝑥 = 𝑥 ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐴))
54abbii 2293 . 2 {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
6 dfnul2 3424 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 2464 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐴} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
85, 6, 73eqtr4i 2208 1 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104   = wceq 1353  wcel 2148  {cab 2163  {crab 2459  c0 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-nul 3423
This theorem is referenced by:  difidALT  3492
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