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Theorem bj-tagex 33547
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 33533 . . 3 (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
2 p0ex 5095 . . . 4 {∅} ∈ V
32biantru 525 . . 3 (sngl 𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
41, 3bitri 267 . 2 (𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
5 unexb 7235 . 2 ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V)
6 df-bj-tag 33535 . . . 4 tag 𝐴 = (sngl 𝐴 ∪ {∅})
76eqcomi 2786 . . 3 (sngl 𝐴 ∪ {∅}) = tag 𝐴
87eleq1i 2849 . 2 ((sngl 𝐴 ∪ {∅}) ∈ V ↔ tag 𝐴 ∈ V)
94, 5, 83bitri 289 1 (𝐴 ∈ V ↔ tag 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  wcel 2106  Vcvv 3397  cun 3789  c0 4140  {csn 4397  sngl bj-csngl 33525  tag bj-ctag 33534
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 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  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-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-pw 4380  df-sn 4398  df-pr 4400  df-uni 4672  df-bj-sngl 33526  df-bj-tag 33535
This theorem is referenced by:  bj-xtagex  33549
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