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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 33798 | . . . . 5 ⊢ (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}) | |
2 | df-rex 3094 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦})) | |
3 | snssi 4616 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴) | |
4 | sseq1 3884 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴)) | |
5 | 4 | biimparc 472 | . . . . . . . 8 ⊢ (({𝑦} ⊆ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴) |
6 | 3, 5 | sylan 572 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴) |
7 | 6 | eximi 1797 | . . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → ∃𝑦 𝑥 ⊆ 𝐴) |
8 | 2, 7 | sylbi 209 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥 ⊆ 𝐴) |
9 | 1, 8 | sylbi 209 | . . . 4 ⊢ (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥 ⊆ 𝐴) |
10 | ax5e 1871 | . . . 4 ⊢ (∃𝑦 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ⊆ 𝐴) |
12 | selpw 4430 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
13 | 11, 12 | sylibr 226 | . 2 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ∈ 𝒫 𝐴) |
14 | 13 | ssriv 3864 | 1 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ∃wrex 3089 ⊆ wss 3831 𝒫 cpw 4423 {csn 4442 sngl bj-csngl 33795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pr 5187 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-rex 3094 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-pw 4425 df-sn 4443 df-pr 4445 df-bj-sngl 33796 |
This theorem is referenced by: bj-snglex 33803 bj-tagss 33810 |
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