Theorem bj-tagss 33280

Description: The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-tagss	⊢ tag 𝐴 ⊆ 𝒫 𝐴

Proof of Theorem bj-tagss

Step	Hyp	Ref	Expression
1		df-bj-tag 33275	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 33270	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5033	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 4991	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4513	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 221	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 3994	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3839	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2157   ∪ cun 3774   ⊆ wss 3776  ∅c0 4123  𝒫 cpw 4358  {csn 4377  sngl bj-csngl 33265  tag bj-ctag 33274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-pw 4360  df-sn 4378  df-pr 4380  df-bj-sngl 33266  df-bj-tag 33275

This theorem is referenced by: (None)