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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eltag | Structured version Visualization version GIF version |
Description: Characterization of the elements of the tagging of a class. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-eltag | ⊢ (𝐴 ∈ tag 𝐵 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-tag 36957 | . . 3 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅}) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ tag 𝐵 ↔ 𝐴 ∈ (sngl 𝐵 ∪ {∅})) |
3 | elun 4162 | . 2 ⊢ (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅})) | |
4 | bj-elsngl 36950 | . . 3 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥}) | |
5 | 0ex 5312 | . . . 4 ⊢ ∅ ∈ V | |
6 | 5 | elsn2 4669 | . . 3 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
7 | 4, 6 | orbi12i 914 | . 2 ⊢ ((𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅}) ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅)) |
8 | 2, 3, 7 | 3bitri 297 | 1 ⊢ (𝐴 ∈ tag 𝐵 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ∪ cun 3960 ∅c0 4338 {csn 4630 sngl bj-csngl 36947 tag bj-ctag 36956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3068 df-v 3479 df-dif 3965 df-un 3967 df-nul 4339 df-sn 4631 df-pr 4633 df-bj-sngl 36948 df-bj-tag 36957 |
This theorem is referenced by: (None) |
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