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Mirrors > Home > MPE Home > Th. List > Mathboxes > unipwrVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of unipwr 41160. (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 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | snid 4594 | . . . 4 ⊢ 𝑥 ∈ {𝑥} |
3 | idn1 40901 | . . . . 5 ⊢ ( 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 ) | |
4 | snelpwi 5328 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
5 | 3, 4 | e1a 40954 | . . . 4 ⊢ ( 𝑥 ∈ 𝐴 ▶ {𝑥} ∈ 𝒫 𝐴 ) |
6 | elunii 4836 | . . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴) | |
7 | 2, 5, 6 | e01an 41019 | . . 3 ⊢ ( 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ ∪ 𝒫 𝐴 ) |
8 | 7 | in1 40898 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴) |
9 | 8 | ssriv 3970 | 1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ⊆ wss 3935 𝒫 cpw 4538 {csn 4560 ∪ cuni 4831 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-pw 4540 df-sn 4561 df-pr 4563 df-uni 4832 df-vd1 40897 |
This theorem is referenced by: (None) |
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