Theorem unissb 4963

Description: Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.) Avoid ax-11 2158. (Revised by BTernaryTau, 28-Dec-2024.)

Assertion

Ref	Expression
unissb	⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unissb

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4934	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
2	1	imbi1i 349	. . . . 5 ⊢ ((𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
3		19.23v 1941	. . . . 5 ⊢ (∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
4	2, 3	bitr4i 278	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
5	4	albii 1817	. . 3 ⊢ (∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
6		elequ1 2115	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
7	6	anbi1d 630	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)))
8		eleq1w 2827	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
9	7, 8	imbi12d 344	. . . . 5 ⊢ (𝑦 = 𝑧 → (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑧 ∈ 𝐵)))
10		elequ2 2123	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
11		eleq1w 2827	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
12	10, 11	anbi12d 631	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴)))
13	12	imbi1d 341	. . . . 5 ⊢ (𝑥 = 𝑧 → (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝐵)))
14	9, 13	alcomw 2044	. . . 4 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
15		19.21v 1938	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
16		impexp 450	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵)))
17		bi2.04 387	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
18	16, 17	bitri 275	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
19	18	albii 1817	. . . . . 6 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
20		df-ss 3993	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
21	20	imbi2i 336	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
22	15, 19, 21	3bitr4i 303	. . . . 5 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
23	22	albii 1817	. . . 4 ⊢ (∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
24	14, 23	bitri 275	. . 3 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
25	5, 24	bitri 275	. 2 ⊢ (∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
26		df-ss 3993	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵))
27		df-ral 3068	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
28	25, 26, 27	3bitr4i 303	1 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  ∪ cuni 4931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-v 3490  df-ss 3993  df-uni 4932

This theorem is referenced by:  uniss2  4965  ssunieq  4967  sspwuni  5123  pwssb  5124  ordunisssuc  6501  sorpssuni  7767  uniordint  7837  sbthlem1  9149  ordunifi  9354  isfinite2  9362  cflim2  10332  fin23lem16  10404  fin23lem29  10410  fin1a2lem11  10479  fin1a2lem13  10481  itunitc  10490  zorng  10573  wuncval2  10816  suplem1pr  11121  suplem2pr  11122  mrcuni  17679  ipodrsfi  18609  mrelatlub  18632  subgint  19190  efgval  19759  toponmre  23122  neips  23142  neiuni  23151  alexsubALTlem2  24077  alexsubALTlem3  24078  tgpconncompeqg  24141  unidmvol  25595  oldf  27914  tglnunirn  28574  uniinn0  32573  elrspunidl  33421  ssdifidllem  33449  ssmxidllem  33466  locfinreflem  33786  zarclsiin  33817  zarclsint  33818  zarcmplem  33827  sxbrsigalem0  34236  dya2iocuni  34248  dya2iocucvr  34249  carsguni  34273  topjoin  36331  fnejoin1  36334  fnejoin2  36335  ovoliunnfl  37622  voliunnfl  37624  volsupnfl  37625  intidl  37989  unichnidl  37991  onuniintrab  43187  onsupmaxb  43200  onsupnub  43210  mnuunid  44246  expanduniss  44262  salexct  46255  unilbss  48549  unilbeu  48657  ipolublem  48658  setrec1lem2  48780  setrec2fun  48784