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

Theorem elneeldif 3962
Description: The elements of a set difference and the minuend are not equal. (Contributed by AV, 21-Oct-2023.)
Assertion
Ref Expression
elneeldif ((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑋𝑌)

Proof of Theorem elneeldif
StepHypRef Expression
1 eldif 3958 . . 3 (𝑌 ∈ (𝐵𝐴) ↔ (𝑌𝐵 ∧ ¬ 𝑌𝐴))
2 nelne2 3039 . . . . 5 ((𝑋𝐴 ∧ ¬ 𝑌𝐴) → 𝑋𝑌)
32ex 412 . . . 4 (𝑋𝐴 → (¬ 𝑌𝐴𝑋𝑌))
43adantld 490 . . 3 (𝑋𝐴 → ((𝑌𝐵 ∧ ¬ 𝑌𝐴) → 𝑋𝑌))
51, 4biimtrid 241 . 2 (𝑋𝐴 → (𝑌 ∈ (𝐵𝐴) → 𝑋𝑌))
65imp 406 1 ((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2105  wne 2939  cdif 3945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-dif 3951
This theorem is referenced by:  frlmsslsp  21571  fmlasucdisj  34689  mhpind  41469
  Copyright terms: Public domain W3C validator