MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difrab Structured version   Visualization version   GIF version

Theorem difrab 4281
Description: Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
difrab ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}

Proof of Theorem difrab
StepHypRef Expression
1 df-rab 3152 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 3152 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2difeq12i 4101 . 2 ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)})
4 df-rab 3152 . . 3 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
5 difab 4276 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓))}
6 anass 469 . . . . . 6 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓)))
7 simpr 485 . . . . . . . . 9 ((𝑥𝐴𝜓) → 𝜓)
87con3i 157 . . . . . . . 8 𝜓 → ¬ (𝑥𝐴𝜓))
98anim2i 616 . . . . . . 7 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) → ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
10 pm3.2 470 . . . . . . . . . 10 (𝑥𝐴 → (𝜓 → (𝑥𝐴𝜓)))
1110adantr 481 . . . . . . . . 9 ((𝑥𝐴𝜑) → (𝜓 → (𝑥𝐴𝜓)))
1211con3d 155 . . . . . . . 8 ((𝑥𝐴𝜑) → (¬ (𝑥𝐴𝜓) → ¬ 𝜓))
1312imdistani 569 . . . . . . 7 (((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)) → ((𝑥𝐴𝜑) ∧ ¬ 𝜓))
149, 13impbii 210 . . . . . 6 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) ↔ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
156, 14bitr3i 278 . . . . 5 ((𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
1615abbii 2891 . . . 4 {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓))}
175, 16eqtr4i 2852 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
184, 17eqtr4i 2852 . 2 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)})
193, 18eqtr4i 2852 1 ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2107  {cab 2804  {crab 3147  cdif 3937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943
This theorem is referenced by:  alephsuc3  9991  shftmbl  24054  musum  25682  clwwlknclwwlkdif  27671  aciunf1  30323  poimirlem26  34785  poimirlem27  34786
  Copyright terms: Public domain W3C validator