Theorem bj-tagci 34420

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-tagci	⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ tag 𝐵)

Proof of Theorem bj-tagci

Step	Hyp	Ref	Expression
1		bj-snglc 34405	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵)
2		bj-sngltagi 34418	. 2 ⊢ ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵)
3	1, 2	sylbi 220	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ tag 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2111  {csn 4525  sngl bj-csngl 34401  tag bj-ctag 34410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-bj-sngl 34402  df-bj-tag 34411

This theorem is referenced by: (None)