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Theorem bj-tagex 33100
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 33086 . . 3 (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
2 p0ex 4883 . . . 4 {∅} ∈ V
32biantru 525 . . 3 (sngl 𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
41, 3bitri 264 . 2 (𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
5 unexb 7000 . 2 ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V)
6 df-bj-tag 33088 . . . 4 tag 𝐴 = (sngl 𝐴 ∪ {∅})
76eqcomi 2660 . . 3 (sngl 𝐴 ∪ {∅}) = tag 𝐴
87eleq1i 2721 . 2 ((sngl 𝐴 ∪ {∅}) ∈ V ↔ tag 𝐴 ∈ V)
94, 5, 83bitri 286 1 (𝐴 ∈ V ↔ tag 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2030  Vcvv 3231  cun 3605  c0 3948  {csn 4210  sngl bj-csngl 33078  tag bj-ctag 33087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213  df-uni 4469  df-bj-sngl 33079  df-bj-tag 33088
This theorem is referenced by:  bj-xtagex  33102
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