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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 4718 | . . . . 5 ⊢ ∅ ∈ V | |
2 | 1 | snid 4155 | . . . 4 ⊢ ∅ ∈ {∅} |
3 | 2 | olci 405 | . . 3 ⊢ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}) |
4 | elun 3715 | . . 3 ⊢ (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})) | |
5 | 3, 4 | mpbir 220 | . 2 ⊢ ∅ ∈ (sngl 𝐴 ∪ {∅}) |
6 | df-bj-tag 32156 | . 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
7 | 5, 6 | eleqtrri 2687 | 1 ⊢ ∅ ∈ tag 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 ∈ wcel 1977 ∪ cun 3538 ∅c0 3874 {csn 4125 sngl bj-csngl 32146 tag bj-ctag 32155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-bj-tag 32156 |
This theorem is referenced by: bj-tagn0 32160 |
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