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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 3703 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴) | |
2 | vex 2636 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | elpw 3455 | . . 3 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴) |
4 | 1, 3 | sylibr 133 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴) |
5 | 4 | ssriv 3043 | 1 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1445 ⊆ wss 3013 𝒫 cpw 3449 ∪ cuni 3675 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-in 3019 df-ss 3026 df-pw 3451 df-uni 3676 |
This theorem is referenced by: uniexb 4323 2pwuninelg 6086 istopon 11864 eltg3i 11908 mopnfss 12233 |
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