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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 36951 | . . . . 5 ⊢ (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}) | |
2 | df-rex 3069 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦})) | |
3 | snssi 4813 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴) | |
4 | sseq1 4021 | . . . . . . . . 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 1832 | . . . . . 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 1910 | . . . 4 ⊢ (∃𝑦 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ⊆ 𝐴) |
12 | velpw 4610 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
13 | 11, 12 | sylibr 234 | . 2 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ∈ 𝒫 𝐴) |
14 | 13 | ssriv 3999 | 1 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 𝒫 cpw 4605 {csn 4631 sngl bj-csngl 36948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rex 3069 df-v 3480 df-un 3968 df-ss 3980 df-pw 4607 df-sn 4632 df-pr 4634 df-bj-sngl 36949 |
This theorem is referenced by: bj-snglex 36956 bj-tagss 36963 |
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