Theorem bj-tagss 33918

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 33913	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 33908	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5154	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5109	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4631	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 231	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4088	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3928	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2083   ∪ cun 3863   ⊆ wss 3865  ∅c0 4217  𝒫 cpw 4459  {csn 4478  sngl bj-csngl 33903  tag bj-ctag 33912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rex 3113  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-pw 4461  df-sn 4479  df-pr 4481  df-bj-sngl 33904  df-bj-tag 33913

This theorem is referenced by: (None)