Theorem bj-tagss 34295

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 34290	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 34285	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5256	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5211	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4718	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 232	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4161	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 4001	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2114   ∪ cun 3934   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4567  sngl bj-csngl 34280  tag bj-ctag 34289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-pw 4541  df-sn 4568  df-pr 4570  df-bj-sngl 34281  df-bj-tag 34290

This theorem is referenced by: (None)