Theorem bj-tagss 36946

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 36941	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 36936	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5374	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5325	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4810	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 230	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4214	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 4043	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2108   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  sngl bj-csngl 36931  tag bj-ctag 36940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-pw 4624  df-sn 4649  df-pr 4651  df-bj-sngl 36932  df-bj-tag 36941

This theorem is referenced by: (None)