Theorem uniprg 4856

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.) Avoid using unipr 4857 to prove it from uniprg 4856. (Revised by BJ, 1-Sep-2024.)

Assertion

Ref	Expression
uniprg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))

Proof of Theorem uniprg

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

Step	Hyp	Ref	Expression
1		vex 3436	. . . . . . . . 9 ⊢ 𝑦 ∈ V
2	1	elpr 4584	. . . . . . . 8 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
3	2	anbi2i 623	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)))
4		ancom 461	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) ∧ 𝑥 ∈ 𝑦))
5		andir 1006	. . . . . . . 8 ⊢ (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) ∧ 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
6	4, 5	bitri 274	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
7	3, 6	bitri 274	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
8	7	exbii 1850	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ∃𝑦((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
9		19.43 1885	. . . . 5 ⊢ (∃𝑦((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)) ↔ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
10	8, 9	bitri 274	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
11		clel3g 3591	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦)))
12	11	bicomd 222	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐴))
13	12	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐴))
14		clel3g 3591	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
15	14	bicomd 222	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐵))
16	15	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐵))
17	13, 16	orbi12d 916	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
18	10, 17	bitrid 282	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
19	18	abbidv 2807	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)})
20		df-uni 4840	. 2 ⊢ ∪ {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})}
21		df-un 3892	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
22	19, 20, 21	3eqtr4g 2803	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ∪ cun 3885  {cpr 4563  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564  df-uni 4840

This theorem is referenced by:  unipr  4857  unisng  4860  wunun  10466  tskun  10542  gruun  10562  mrcun  17331  unopn  22052  indistopon  22151  unconn  22580  limcun  25059  sshjval3  29716  prsiga  32099  unelsiga  32102  unelldsys  32126  measxun2  32178  measssd  32183  carsgsigalem  32282  carsgclctun  32288  pmeasmono  32291  probun  32386  indispconn  33196  bj-prmoore  35286  kelac2  40890  mnuund  41896  fourierdlem70  43717  fourierdlem71  43718  saluncl  43858  prsal  43859  meadjun  44000  omeunle  44054  toplatjoin  46288