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Theorem bj-tagex 36970
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 36956 . . 3 (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
2 p0ex 5390 . . . 4 {∅} ∈ V
32biantru 529 . . 3 (sngl 𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
41, 3bitri 275 . 2 (𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
5 unexb 7766 . 2 ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V)
6 df-bj-tag 36958 . . . 4 tag 𝐴 = (sngl 𝐴 ∪ {∅})
76eqcomi 2744 . . 3 (sngl 𝐴 ∪ {∅}) = tag 𝐴
87eleq1i 2830 . 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 206  wa 395  wcel 2106  Vcvv 3478  cun 3961  c0 4339  {csn 4631  sngl bj-csngl 36948  tag bj-ctag 36957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-bj-sngl 36949  df-bj-tag 36958
This theorem is referenced by:  bj-xtagex  36972
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