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Theorem bj-tagex 33252
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 33238 . . 3 (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
2 p0ex 4990 . . . 4 {∅} ∈ V
32biantru 527 . . 3 (sngl 𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
41, 3bitri 264 . 2 (𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
5 unexb 7111 . 2 ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V)
6 df-bj-tag 33240 . . . 4 tag 𝐴 = (sngl 𝐴 ∪ {∅})
76eqcomi 2757 . . 3 (sngl 𝐴 ∪ {∅}) = tag 𝐴
87eleq1i 2818 . 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 2127  Vcvv 3328  cun 3701  c0 4046  {csn 4309  sngl bj-csngl 33230  tag bj-ctag 33239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-pw 4292  df-sn 4310  df-pr 4312  df-uni 4577  df-bj-sngl 33231  df-bj-tag 33240
This theorem is referenced by:  bj-xtagex  33254
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