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Theorem difsnexi 7467
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 488 . . . . 5 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∖ {𝐾}) ∈ V)
2 snex 5300 . . . . 5 {𝐾} ∈ V
3 unexg 7456 . . . . 5 (((𝑁 ∖ {𝐾}) ∈ V ∧ {𝐾} ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)
41, 2, 3sylancl 589 . . . 4 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)
5 difsnid 4706 . . . . . . 7 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
65eqcomd 2807 . . . . . 6 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
76eleq1d 2877 . . . . 5 (𝐾𝑁 → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V))
87adantr 484 . . . 4 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V))
94, 8mpbird 260 . . 3 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → 𝑁 ∈ V)
109ex 416 . 2 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V))
11 difsn 4694 . . . 4 𝐾𝑁 → (𝑁 ∖ {𝐾}) = 𝑁)
1211eleq1d 2877 . . 3 𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V ↔ 𝑁 ∈ V))
1312biimpd 232 . 2 𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V))
1410, 13pm2.61i 185 1 ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2112  Vcvv 3444  cdif 3881  cun 3882  {csn 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-uni 4804
This theorem is referenced by:  pmtrdifellem1  18600  pmtrdifellem2  18601  tgdif0  21601
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