Theorem bj-tagcg 37716

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 37700	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵)
2		bj-sngltag 37714	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))
3	1, 2	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ tag 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145  {csn 4587  sngl bj-csngl 37696  tag bj-ctag 37705

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-sn 4588  df-pr 4590  df-bj-sngl 37697  df-bj-tag 37706

This theorem is used by: (None)