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

Theorem difsnexi 6934
Description: If the difference of a class and a singleton is a set, the class itself is a set. (Contributed by AV, 15-Jan-2019.)
Assertion
Ref Expression
difsnexi ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)

Proof of Theorem difsnexi
StepHypRef Expression
1 simpr 477 . . . . 5 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∖ {𝐾}) ∈ V)
2 snex 4879 . . . . 5 {𝐾} ∈ V
3 unexg 6924 . . . . 5 (((𝑁 ∖ {𝐾}) ∈ V ∧ {𝐾} ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)
41, 2, 3sylancl 693 . . . 4 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)
5 difsnid 4317 . . . . . . 7 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
65eqcomd 2627 . . . . . 6 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
76eleq1d 2683 . . . . 5 (𝐾𝑁 → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V))
87adantr 481 . . . 4 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V))
94, 8mpbird 247 . . 3 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → 𝑁 ∈ V)
109ex 450 . 2 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V))
11 difsn 4304 . . . 4 𝐾𝑁 → (𝑁 ∖ {𝐾}) = 𝑁)
1211eleq1d 2683 . . 3 𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V ↔ 𝑁 ∈ V))
1312biimpd 219 . 2 𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V))
1410, 13pm2.61i 176 1 ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wcel 1987  Vcvv 3190  cdif 3557  cun 3558  {csn 4155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-sn 4156  df-pr 4158  df-uni 4410
This theorem is referenced by:  pmtrdifellem1  17836  pmtrdifellem2  17837  tgdif0  20736
  Copyright terms: Public domain W3C validator