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Mirrors > Home > ILE Home > Th. List > pwuni | GIF version |
Description: A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) |
Ref | Expression |
---|---|
pwuni | ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 3759 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴) | |
2 | vex 2684 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | elpw 3511 | . . 3 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴) |
4 | 1, 3 | sylibr 133 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴) |
5 | 4 | ssriv 3096 | 1 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 ⊆ wss 3066 𝒫 cpw 3505 ∪ cuni 3731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-pw 3507 df-uni 3732 |
This theorem is referenced by: uniexb 4389 2pwuninelg 6173 istopon 12169 eltg3i 12214 mopnfss 12605 |
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