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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sngltag | Structured version Visualization version GIF version | ||
| Description: The singletonization and the tagging of a set contain the same singletons. (Contributed by BJ, 6-Oct-2018.) |
| Ref | Expression |
|---|---|
| bj-sngltag | ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-sngltagi 36983 | . 2 ⊢ ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵) | |
| 2 | df-bj-tag 36976 | . . . 4 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅}) | |
| 3 | 2 | eleq2i 2833 | . . 3 ⊢ ({𝐴} ∈ tag 𝐵 ↔ {𝐴} ∈ (sngl 𝐵 ∪ {∅})) |
| 4 | elun 4153 | . . . 4 ⊢ ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) ↔ ({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅})) | |
| 5 | idd 24 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ sngl 𝐵)) | |
| 6 | elsni 4643 | . . . . . 6 ⊢ ({𝐴} ∈ {∅} → {𝐴} = ∅) | |
| 7 | snprc 4717 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 8 | elex 3501 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
| 9 | 8 | pm2.24d 151 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ V → {𝐴} ∈ sngl 𝐵)) |
| 10 | 7, 9 | biimtrrid 243 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → {𝐴} ∈ sngl 𝐵)) |
| 11 | 6, 10 | syl5 34 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ {∅} → {𝐴} ∈ sngl 𝐵)) |
| 12 | 5, 11 | jaod 860 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}) → {𝐴} ∈ sngl 𝐵)) |
| 13 | 4, 12 | biimtrid 242 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) → {𝐴} ∈ sngl 𝐵)) |
| 14 | 3, 13 | biimtrid 242 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ tag 𝐵 → {𝐴} ∈ sngl 𝐵)) |
| 15 | 1, 14 | impbid2 226 | 1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ 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 |
| 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-ss 3968 df-nul 4334 df-sn 4627 df-bj-tag 36976 |
| This theorem is referenced by: bj-tagcg 36986 bj-taginv 36987 |
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