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Theorem difsnexi 7247
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 479 . . . . 5 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∖ {𝐾}) ∈ V)
2 snex 5140 . . . . 5 {𝐾} ∈ V
3 unexg 7236 . . . . 5 (((𝑁 ∖ {𝐾}) ∈ V ∧ {𝐾} ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)
41, 2, 3sylancl 580 . . . 4 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)
5 difsnid 4572 . . . . . . 7 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
65eqcomd 2784 . . . . . 6 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
76eleq1d 2844 . . . . 5 (𝐾𝑁 → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V))
87adantr 474 . . . 4 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V))
94, 8mpbird 249 . . 3 ((𝐾𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → 𝑁 ∈ V)
109ex 403 . 2 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V))
11 difsn 4560 . . . 4 𝐾𝑁 → (𝑁 ∖ {𝐾}) = 𝑁)
1211eleq1d 2844 . . 3 𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V ↔ 𝑁 ∈ V))
1312biimpd 221 . 2 𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V))
1410, 13pm2.61i 177 1 ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wcel 2107  Vcvv 3398  cdif 3789  cun 3790  {csn 4398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-sn 4399  df-pr 4401  df-uni 4672
This theorem is referenced by:  pmtrdifellem1  18279  pmtrdifellem2  18280  tgdif0  21204
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