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Mirrors > Home > MPE Home > Th. List > unissb | Structured version Visualization version GIF version |
Description: Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.) |
Ref | Expression |
---|---|
unissb | ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni 4834 | . . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) | |
2 | 1 | imbi1i 352 | . . . . 5 ⊢ ((𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)) |
3 | 19.23v 1939 | . . . . 5 ⊢ (∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)) | |
4 | 2, 3 | bitr4i 280 | . . . 4 ⊢ ((𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)) |
5 | 4 | albii 1816 | . . 3 ⊢ (∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)) |
6 | alcom 2159 | . . . 4 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)) | |
7 | 19.21v 1936 | . . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))) | |
8 | impexp 453 | . . . . . . . 8 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵))) | |
9 | bi2.04 391 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))) | |
10 | 8, 9 | bitri 277 | . . . . . . 7 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))) |
11 | 10 | albii 1816 | . . . . . 6 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))) |
12 | dfss2 3954 | . . . . . . 7 ⊢ (𝑥 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)) | |
13 | 12 | imbi2i 338 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))) |
14 | 7, 11, 13 | 3bitr4i 305 | . . . . 5 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵)) |
15 | 14 | albii 1816 | . . . 4 ⊢ (∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵)) |
16 | 6, 15 | bitri 277 | . . 3 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵)) |
17 | 5, 16 | bitri 277 | . 2 ⊢ (∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵)) |
18 | dfss2 3954 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵)) | |
19 | df-ral 3143 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵)) | |
20 | 17, 18, 19 | 3bitr4i 305 | 1 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 ∃wex 1776 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ∪ cuni 4831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-in 3942 df-ss 3951 df-uni 4832 |
This theorem is referenced by: uniss2 4863 ssunieq 4865 sspwuni 5014 pwssb 5015 ordunisssuc 6287 sorpssuni 7452 uniordint 7515 sbthlem1 8621 ordunifi 8762 isfinite2 8770 cflim2 9679 fin23lem16 9751 fin23lem29 9757 fin1a2lem11 9826 fin1a2lem13 9828 itunitc 9837 zorng 9920 wuncval2 10163 suplem1pr 10468 suplem2pr 10469 mrcuni 16886 ipodrsfi 17767 mrelatlub 17790 subgint 18297 efgval 18837 toponmre 21695 neips 21715 neiuni 21724 alexsubALTlem2 22650 alexsubALTlem3 22651 tgpconncompeqg 22714 unidmvol 24136 tglnunirn 26328 uniinn0 30296 ssmxidllem 30973 locfinreflem 31099 sxbrsigalem0 31524 dya2iocuni 31536 dya2iocucvr 31537 carsguni 31561 topjoin 33708 fnejoin1 33711 fnejoin2 33712 ovoliunnfl 34928 voliunnfl 34930 volsupnfl 34931 intidl 35301 unichnidl 35303 mnuunid 40606 expanduniss 40622 salexct 42611 setrec1lem2 44785 setrec2fun 44789 |
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