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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-snglss | Structured version Visualization version GIF version | ||
| Description: The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.) |
| Ref | Expression |
|---|---|
| bj-snglss | ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-elsngl 36969 | . . . . 5 ⊢ (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}) | |
| 2 | df-rex 3071 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦})) | |
| 3 | snssi 4808 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴) | |
| 4 | sseq1 4009 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴)) | |
| 5 | 4 | biimparc 479 | . . . . . . . 8 ⊢ (({𝑦} ⊆ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴) |
| 6 | 3, 5 | sylan 580 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴) |
| 7 | 6 | eximi 1835 | . . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → ∃𝑦 𝑥 ⊆ 𝐴) |
| 8 | 2, 7 | sylbi 217 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥 ⊆ 𝐴) |
| 9 | 1, 8 | sylbi 217 | . . . 4 ⊢ (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥 ⊆ 𝐴) |
| 10 | ax5e 1912 | . . . 4 ⊢ (∃𝑦 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴) | |
| 11 | 9, 10 | syl 17 | . . 3 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ⊆ 𝐴) |
| 12 | velpw 4605 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
| 13 | 11, 12 | sylibr 234 | . 2 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ∈ 𝒫 𝐴) |
| 14 | 13 | ssriv 3987 | 1 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 sngl bj-csngl 36966 |
| 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-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-v 3482 df-un 3956 df-ss 3968 df-pw 4602 df-sn 4627 df-pr 4629 df-bj-sngl 36967 |
| This theorem is referenced by: bj-snglex 36974 bj-tagss 36981 |
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