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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-tagss | Structured version Visualization version GIF version |
Description: The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-tagss | ⊢ tag 𝐴 ⊆ 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-tag 36582 | . 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
2 | bj-snglss 36577 | . . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 | |
3 | 0elpw 5356 | . . . 4 ⊢ ∅ ∈ 𝒫 𝐴 | |
4 | 0ex 5308 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 4 | snss 4791 | . . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴) |
6 | 3, 5 | mpbi 229 | . . 3 ⊢ {∅} ⊆ 𝒫 𝐴 |
7 | 2, 6 | unssi 4183 | . 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴 |
8 | 1, 7 | eqsstri 4011 | 1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∪ cun 3942 ⊆ wss 3944 ∅c0 4322 𝒫 cpw 4604 {csn 4630 sngl bj-csngl 36572 tag bj-ctag 36581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rex 3060 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-pw 4606 df-sn 4631 df-pr 4633 df-bj-sngl 36573 df-bj-tag 36582 |
This theorem is referenced by: (None) |
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