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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 3822 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴) | |
2 | vex 2733 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | elpw 3570 | . . 3 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴) |
4 | 1, 3 | sylibr 133 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴) |
5 | 4 | ssriv 3151 | 1 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 ⊆ wss 3121 𝒫 cpw 3564 ∪ cuni 3794 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-pw 3566 df-uni 3795 |
This theorem is referenced by: uniexb 4456 2pwuninelg 6259 istopon 12764 eltg3i 12809 mopnfss 13200 |
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