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Theorem bj-tagex 36324
Description: A class is a set if and only if its tagging is a set. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-tagex (𝐴 ∈ V ↔ tag 𝐴 ∈ V)

Proof of Theorem bj-tagex
StepHypRef Expression
1 bj-snglex 36310 . . 3 (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
2 p0ex 5372 . . . 4 {∅} ∈ V
32biantru 529 . . 3 (sngl 𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
41, 3bitri 275 . 2 (𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
5 unexb 7728 . 2 ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V)
6 df-bj-tag 36312 . . . 4 tag 𝐴 = (sngl 𝐴 ∪ {∅})
76eqcomi 2733 . . 3 (sngl 𝐴 ∪ {∅}) = tag 𝐴
87eleq1i 2816 . 2 ((sngl 𝐴 ∪ {∅}) ∈ V ↔ tag 𝐴 ∈ V)
94, 5, 83bitri 297 1 (𝐴 ∈ V ↔ tag 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2098  Vcvv 3466  cun 3938  c0 4314  {csn 4620  sngl bj-csngl 36302  tag bj-ctag 36311
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-pw 4596  df-sn 4621  df-pr 4623  df-uni 4900  df-bj-sngl 36303  df-bj-tag 36312
This theorem is referenced by:  bj-xtagex  36326
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