| Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
| 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 36976 | . 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
| 2 | bj-snglss 36971 | . . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 | |
| 3 | 0elpw 5356 | . . . 4 ⊢ ∅ ∈ 𝒫 𝐴 | |
| 4 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 5 | 4 | snss 4785 | . . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴) |
| 6 | 3, 5 | mpbi 230 | . . 3 ⊢ {∅} ⊆ 𝒫 𝐴 |
| 7 | 2, 6 | unssi 4191 | . 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴 |
| 8 | 1, 7 | eqsstri 4030 | 1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 sngl bj-csngl 36966 tag bj-ctag 36975 |
| 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 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-bj-sngl 36967 df-bj-tag 36976 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |