Theorem bj-tagss 37466

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 37461	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 37456	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5313	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5258	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4744	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 232	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4144	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3983	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2143   ∪ cun 3903   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4556  {csn 4583  sngl bj-csngl 37451  tag bj-ctag 37460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3088  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-pw 4558  df-sn 4584  df-pr 4586  df-bj-sngl 37452  df-bj-tag 37461

This theorem is referenced by: (None)