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Theorem difsnexi 7745
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 484 . . . . 5 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∖ {𝐾}) ∈ V)
2 snex 5424 . . . . 5 {𝐾} ∈ V
3 unexg 7733 . . . . 5 (((𝑁 ∖ {𝐾}) ∈ V ∧ {𝐾} ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)
41, 2, 3sylancl 585 . . . 4 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)
5 difsnid 4808 . . . . . . 7 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
65eqcomd 2732 . . . . . 6 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
76eleq1d 2812 . . . . 5 (𝐾𝑁 → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V))
87adantr 480 . . . 4 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V))
94, 8mpbird 257 . . 3 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → 𝑁 ∈ V)
109ex 412 . 2 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V))
11 difsn 4796 . . . 4 𝐾𝑁 → (𝑁 ∖ {𝐾}) = 𝑁)
1211eleq1d 2812 . . 3 𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V ↔ 𝑁 ∈ V))
1312biimpd 228 . 2 𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V))
1410, 13pm2.61i 182 1 ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wcel 2098  Vcvv 3468  cdif 3940  cun 3941  {csn 4623
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-uni 4903
This theorem is referenced by:  pmtrdifellem1  19396  pmtrdifellem2  19397  tgdif0  22850
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