Theorem bj-tagcg 35175

Description: Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-tagcg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ tag 𝐵))

Proof of Theorem bj-tagcg

Step	Hyp	Ref	Expression
1		bj-snglc 35159	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵)
2		bj-sngltag 35173	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))
3	1, 2	syl5bb 283	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ tag 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106  {csn 4561  sngl bj-csngl 35155  tag bj-ctag 35164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-pr 4564  df-bj-sngl 35156  df-bj-tag 35165

This theorem is referenced by: (None)