| Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-0eltag | Structured version Visualization version GIF version | ||
| Description: The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.) |
| Ref | Expression |
|---|---|
| bj-0eltag | ⊢ ∅ ∈ tag 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 2 | 1 | snid 4662 | . . . 4 ⊢ ∅ ∈ {∅} |
| 3 | 2 | olci 867 | . . 3 ⊢ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}) |
| 4 | elun 4153 | . . 3 ⊢ (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})) | |
| 5 | 3, 4 | mpbir 231 | . 2 ⊢ ∅ ∈ (sngl 𝐴 ∪ {∅}) |
| 6 | df-bj-tag 36976 | . 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
| 7 | 5, 6 | eleqtrri 2840 | 1 ⊢ ∅ ∈ tag 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∨ wo 848 ∈ wcel 2108 ∪ cun 3949 ∅c0 4333 {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-ext 2708 ax-nul 5306 |
| 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-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-bj-tag 36976 |
| This theorem is referenced by: bj-tagn0 36980 |
| Copyright terms: Public domain | W3C validator |