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Mirrors > Home > ILE Home > Th. List > iunpwss | GIF version |
Description: Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) |
Ref | Expression |
---|---|
iunpwss | ⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssiun 3802 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥) | |
2 | eliun 3764 | . . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥) | |
3 | vex 2644 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 3 | elpw 3463 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥) |
5 | 4 | rexbii 2401 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥) |
6 | 2, 5 | bitri 183 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥) |
7 | 3 | elpw 3463 | . . . 4 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴) |
8 | uniiun 3813 | . . . . 5 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
9 | 8 | sseq2i 3074 | . . . 4 ⊢ (𝑦 ⊆ ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥) |
10 | 7, 9 | bitri 183 | . . 3 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥) |
11 | 1, 6, 10 | 3imtr4i 200 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → 𝑦 ∈ 𝒫 ∪ 𝐴) |
12 | 11 | ssriv 3051 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 ∃wrex 2376 ⊆ wss 3021 𝒫 cpw 3457 ∪ cuni 3683 ∪ ciun 3760 |
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 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-in 3027 df-ss 3034 df-pw 3459 df-uni 3684 df-iun 3762 |
This theorem is referenced by: (None) |
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