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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 33913 | . 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
2 | bj-snglss 33908 | . . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 | |
3 | 0elpw 5154 | . . . 4 ⊢ ∅ ∈ 𝒫 𝐴 | |
4 | 0ex 5109 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 4 | snss 4631 | . . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴) |
6 | 3, 5 | mpbi 231 | . . 3 ⊢ {∅} ⊆ 𝒫 𝐴 |
7 | 2, 6 | unssi 4088 | . 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴 |
8 | 1, 7 | eqsstri 3928 | 1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2083 ∪ cun 3863 ⊆ wss 3865 ∅c0 4217 𝒫 cpw 4459 {csn 4478 sngl bj-csngl 33903 tag bj-ctag 33912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-rex 3113 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-pw 4461 df-sn 4479 df-pr 4481 df-bj-sngl 33904 df-bj-tag 33913 |
This theorem is referenced by: (None) |
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