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Mirrors > Home > MPE Home > Th. List > Mathboxes > unipwrVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of unipwr 44170. (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 3472 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | snid 4659 | . . . 4 ⊢ 𝑥 ∈ {𝑥} |
3 | idn1 43911 | . . . . 5 ⊢ ( 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 ) | |
4 | snelpwi 5436 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
5 | 3, 4 | e1a 43964 | . . . 4 ⊢ ( 𝑥 ∈ 𝐴 ▶ {𝑥} ∈ 𝒫 𝐴 ) |
6 | elunii 4907 | . . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴) | |
7 | 2, 5, 6 | e01an 44029 | . . 3 ⊢ ( 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ ∪ 𝒫 𝐴 ) |
8 | 7 | in1 43908 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴) |
9 | 8 | ssriv 3981 | 1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ⊆ wss 3943 𝒫 cpw 4597 {csn 4623 ∪ cuni 4902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-un 3948 df-in 3950 df-ss 3960 df-pw 4599 df-sn 4624 df-pr 4626 df-uni 4903 df-vd1 43907 |
This theorem is referenced by: (None) |
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