Theorem bj-tagss 36941

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 36936	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 36931	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5306	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5257	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4745	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 230	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4150	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3990	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2109   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585  sngl bj-csngl 36926  tag bj-ctag 36935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-pw 4561  df-sn 4586  df-pr 4588  df-bj-sngl 36927  df-bj-tag 36936

This theorem is referenced by: (None)