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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 34844 | . . . . 5 ⊢ (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}) | |
2 | df-rex 3057 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦})) | |
3 | snssi 4707 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴) | |
4 | sseq1 3912 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴)) | |
5 | 4 | biimparc 483 | . . . . . . . 8 ⊢ (({𝑦} ⊆ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴) |
6 | 3, 5 | sylan 583 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴) |
7 | 6 | eximi 1842 | . . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → ∃𝑦 𝑥 ⊆ 𝐴) |
8 | 2, 7 | sylbi 220 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥 ⊆ 𝐴) |
9 | 1, 8 | sylbi 220 | . . . 4 ⊢ (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥 ⊆ 𝐴) |
10 | ax5e 1920 | . . . 4 ⊢ (∃𝑦 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ⊆ 𝐴) |
12 | velpw 4504 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
13 | 11, 12 | sylibr 237 | . 2 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ∈ 𝒫 𝐴) |
14 | 13 | ssriv 3891 | 1 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ∃wrex 3052 ⊆ wss 3853 𝒫 cpw 4499 {csn 4527 sngl bj-csngl 34841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rex 3057 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-pw 4501 df-sn 4528 df-pr 4530 df-bj-sngl 34842 |
This theorem is referenced by: bj-snglex 34849 bj-tagss 34856 |
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