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Theorem dfnul3 3224
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 1589 . . . . 5 𝑥 = 𝑥
21notnoti 574 . . . 4 ¬ ¬ 𝑥 = 𝑥
3 pm3.24 627 . . . 4 ¬ (𝑥𝐴 ∧ ¬ 𝑥𝐴)
42, 32false 617 . . 3 𝑥 = 𝑥 ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐴))
54abbii 2153 . 2 {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
6 dfnul2 3223 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 2312 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐴} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
85, 6, 73eqtr4i 2070 1 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 97   = wceq 1243  wcel 1393  {cab 2026  {crab 2307  c0 3221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2312  df-v 2556  df-dif 2917  df-nul 3222
This theorem is referenced by:  difidALT  3290
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