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Mirrors > Home > NFE 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 ℘x ⊆ ℘∪A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssiun 4008 | . . 3 ⊢ (∃x ∈ A y ⊆ x → y ⊆ ∪x ∈ A x) | |
2 | eliun 3973 | . . . 4 ⊢ (y ∈ ∪x ∈ A ℘x ↔ ∃x ∈ A y ∈ ℘x) | |
3 | vex 2862 | . . . . . 6 ⊢ y ∈ V | |
4 | 3 | elpw 3728 | . . . . 5 ⊢ (y ∈ ℘x ↔ y ⊆ x) |
5 | 4 | rexbii 2639 | . . . 4 ⊢ (∃x ∈ A y ∈ ℘x ↔ ∃x ∈ A y ⊆ x) |
6 | 2, 5 | bitri 240 | . . 3 ⊢ (y ∈ ∪x ∈ A ℘x ↔ ∃x ∈ A y ⊆ x) |
7 | 3 | elpw 3728 | . . . 4 ⊢ (y ∈ ℘∪A ↔ y ⊆ ∪A) |
8 | uniiun 4019 | . . . . 5 ⊢ ∪A = ∪x ∈ A x | |
9 | 8 | sseq2i 3296 | . . . 4 ⊢ (y ⊆ ∪A ↔ y ⊆ ∪x ∈ A x) |
10 | 7, 9 | bitri 240 | . . 3 ⊢ (y ∈ ℘∪A ↔ y ⊆ ∪x ∈ A x) |
11 | 1, 6, 10 | 3imtr4i 257 | . 2 ⊢ (y ∈ ∪x ∈ A ℘x → y ∈ ℘∪A) |
12 | 11 | ssriv 3277 | 1 ⊢ ∪x ∈ A ℘x ⊆ ℘∪A |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 ∃wrex 2615 ⊆ wss 3257 ℘cpw 3722 ∪cuni 3891 ∪ciun 3969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-pw 3724 df-uni 3892 df-iun 3971 |
This theorem is referenced by: (None) |
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