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Theorem difsnexi 7117
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 471 . . . . 5 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∖ {𝐾}) ∈ V)
2 snex 5036 . . . . 5 {𝐾} ∈ V
3 unexg 7106 . . . . 5 (((𝑁 ∖ {𝐾}) ∈ V ∧ {𝐾} ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)
41, 2, 3sylancl 574 . . . 4 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)
5 difsnid 4476 . . . . . . 7 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
65eqcomd 2777 . . . . . 6 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
76eleq1d 2835 . . . . 5 (𝐾𝑁 → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V))
87adantr 466 . . . 4 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V))
94, 8mpbird 247 . . 3 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → 𝑁 ∈ V)
109ex 397 . 2 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V))
11 difsn 4464 . . . 4 𝐾𝑁 → (𝑁 ∖ {𝐾}) = 𝑁)
1211eleq1d 2835 . . 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 382  wcel 2145  Vcvv 3351  cdif 3720  cun 3721  {csn 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-sn 4317  df-pr 4319  df-uni 4575
This theorem is referenced by:  pmtrdifellem1  18103  pmtrdifellem2  18104  tgdif0  21017
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