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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 36976 | . . 3 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅}) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ tag 𝐵 ↔ 𝐴 ∈ (sngl 𝐵 ∪ {∅})) |
| 3 | elun 4153 | . 2 ⊢ (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅})) | |
| 4 | bj-elsngl 36969 | . . 3 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥}) | |
| 5 | 0ex 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 6 | 5 | elsn2 4665 | . . 3 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
| 7 | 4, 6 | orbi12i 915 | . 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 848 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∪ 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-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-rex 3071 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 df-bj-sngl 36967 df-bj-tag 36976 |
| This theorem is referenced by: (None) |
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