Theorem unissb 4944

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 2152. (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 4912	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
2	1	imbi1i 348	. . . . 5 ⊢ ((𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
3		19.23v 1943	. . . . 5 ⊢ (∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
4	2, 3	bitr4i 277	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
5	4	albii 1819	. . 3 ⊢ (∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
6		elequ1 2111	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
7	6	anbi1d 628	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)))
8		eleq1w 2814	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
9	7, 8	imbi12d 343	. . . . 5 ⊢ (𝑦 = 𝑧 → (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑧 ∈ 𝐵)))
10		elequ2 2119	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
11		eleq1w 2814	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
12	10, 11	anbi12d 629	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴)))
13	12	imbi1d 340	. . . . 5 ⊢ (𝑥 = 𝑧 → (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝐵)))
14	9, 13	alcomw 2045	. . . 4 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
15		19.21v 1940	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
16		impexp 449	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵)))
17		bi2.04 386	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
18	16, 17	bitri 274	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
19	18	albii 1819	. . . . . 6 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
20		dfss2 3969	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
21	20	imbi2i 335	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
22	15, 19, 21	3bitr4i 302	. . . . 5 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
23	22	albii 1819	. . . 4 ⊢ (∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
24	14, 23	bitri 274	. . 3 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
25	5, 24	bitri 274	. 2 ⊢ (∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
26		dfss2 3969	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵))
27		df-ral 3060	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
28	25, 26, 27	3bitr4i 302	1 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1537  ∃wex 1779   ∈ wcel 2104  ∀wral 3059   ⊆ wss 3949  ∪ cuni 4909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-v 3474  df-in 3956  df-ss 3966  df-uni 4910

This theorem is referenced by:  uniss2  4946  ssunieq  4948  sspwuni  5104  pwssb  5105  ordunisssuc  6471  sorpssuni  7726  uniordint  7793  sbthlem1  9087  ordunifi  9297  isfinite2  9305  cflim2  10262  fin23lem16  10334  fin23lem29  10340  fin1a2lem11  10409  fin1a2lem13  10411  itunitc  10420  zorng  10503  wuncval2  10746  suplem1pr  11051  suplem2pr  11052  mrcuni  17571  ipodrsfi  18498  mrelatlub  18521  subgint  19068  efgval  19628  toponmre  22819  neips  22839  neiuni  22848  alexsubALTlem2  23774  alexsubALTlem3  23775  tgpconncompeqg  23838  unidmvol  25292  oldf  27587  tglnunirn  28064  uniinn0  32047  elrspunidl  32818  ssmxidllem  32861  locfinreflem  33116  zarclsiin  33147  zarclsint  33148  zarcmplem  33157  sxbrsigalem0  33566  dya2iocuni  33578  dya2iocucvr  33579  carsguni  33603  topjoin  35555  fnejoin1  35558  fnejoin2  35559  ovoliunnfl  36835  voliunnfl  36837  volsupnfl  36838  intidl  37202  unichnidl  37204  onuniintrab  42279  onsupmaxb  42292  onsupnub  42302  mnuunid  43340  expanduniss  43356  salexct  45350  unilbss  47591  unilbeu  47699  ipolublem  47700  setrec1lem2  47822  setrec2fun  47826