Theorem bj-tagex 36375

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

Step	Hyp	Ref	Expression
1		bj-snglex 36361	. . 3 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
2		p0ex 5375	. . . 4 ⊢ {∅} ∈ V
3	2	biantru 529	. . 3 ⊢ (sngl 𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
4	1, 3	bitri 275	. 2 ⊢ (𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V))
5		unexb 7732	. 2 ⊢ ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V)
6		df-bj-tag 36363	. . . 4 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
7	6	eqcomi 2735	. . 3 ⊢ (sngl 𝐴 ∪ {∅}) = tag 𝐴
8	7	eleq1i 2818	. 2 ⊢ ((sngl 𝐴 ∪ {∅}) ∈ V ↔ tag 𝐴 ∈ V)
9	4, 5, 8	3bitri 297	1 ⊢ (𝐴 ∈ V ↔ tag 𝐴 ∈ V)

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2098  Vcvv 3468   ∪ cun 3941  ∅c0 4317  {csn 4623  sngl bj-csngl 36353  tag bj-ctag 36362

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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

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 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-pw 4599  df-sn 4624  df-pr 4626  df-uni 4903  df-bj-sngl 36354  df-bj-tag 36363

This theorem is referenced by:  bj-xtagex  36377