![]() |
Mathbox for Alan Sare |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > unipwrVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of unipwr 39818. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
unipwrVD | ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3387 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | snid 4399 | . . . 4 ⊢ 𝑥 ∈ {𝑥} |
3 | idn1 39549 | . . . . 5 ⊢ ( 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 ) | |
4 | snelpwi 5102 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
5 | 3, 4 | e1a 39611 | . . . 4 ⊢ ( 𝑥 ∈ 𝐴 ▶ {𝑥} ∈ 𝒫 𝐴 ) |
6 | elunii 4632 | . . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴) | |
7 | 2, 5, 6 | e01an 39676 | . . 3 ⊢ ( 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ ∪ 𝒫 𝐴 ) |
8 | 7 | in1 39546 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴) |
9 | 8 | ssriv 3801 | 1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 ⊆ wss 3768 𝒫 cpw 4348 {csn 4367 ∪ cuni 4627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pr 5096 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-v 3386 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-nul 4115 df-pw 4350 df-sn 4368 df-pr 4370 df-uni 4628 df-vd1 39545 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |