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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 5197 | . . . . 5 ⊢ ∅ ∈ V | |
2 | 1 | snid 4587 | . . . 4 ⊢ ∅ ∈ {∅} |
3 | 2 | olci 862 | . . 3 ⊢ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}) |
4 | elun 4113 | . . 3 ⊢ (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})) | |
5 | 3, 4 | mpbir 233 | . 2 ⊢ ∅ ∈ (sngl 𝐴 ∪ {∅}) |
6 | df-bj-tag 34303 | . 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
7 | 5, 6 | eleqtrri 2912 | 1 ⊢ ∅ ∈ tag 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 ∈ wcel 2114 ∪ cun 3922 ∅c0 4279 {csn 4553 sngl bj-csngl 34293 tag bj-ctag 34302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5196 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3488 df-dif 3927 df-un 3929 df-nul 4280 df-sn 4554 df-bj-tag 34303 |
This theorem is referenced by: bj-tagn0 34307 |
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