Theorem uniprg 4855

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 4856 to prove it from uniprg 4855. (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 3435	. . . . . . . . 9 ⊢ 𝑦 ∈ V
2	1	elpr 4581	. . . . . . . 8 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
3	2	anbi2i 629	. . . . . . 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 1016	. . . . . . . 8 ⊢ (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) ∧ 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
6	4, 5	bitri 276	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
7	3, 6	bitri 276	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
8	7	exbii 1855	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ∃𝑦((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
9		19.43 1889	. . . . 5 ⊢ (∃𝑦((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)) ↔ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
10	8, 9	bitri 276	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
11		clel3g 3599	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦)))
12	11	bicomd 224	. . . . . 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 3599	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
15	14	bicomd 224	. . . . . 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 924	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
18	10, 17	bitrid 284	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
19	18	abbidv 2805	. 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 3888	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
22	19, 20, 21	3eqtr4g 2799	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717   ∪ cun 3881  {cpr 4558  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-un 3888  df-sn 4557  df-pr 4559  df-uni 4840

This theorem is referenced by:  unipr  4856  unisng  4857  unexg  7687  wunun  10625  tskun  10701  gruun  10721  mrcun  17580  unopn  22887  indistopon  22985  unconn  23413  limcun  25881  sshjval3  31444  prsiga  34324  unelsiga  34327  unelldsys  34351  measxun2  34403  measssd  34408  carsgsigalem  34508  carsgclctun  34514  pmeasmono  34517  probun  34612  indispconn  35471  bj-prmoore  37482  kelac2  43519  onsucunipr  43826  onsucunitp  43827  oaun2  43835  oaun3  43836  mnuund  44731  fourierdlem70  46627  fourierdlem71  46628  saluncl  46768  prsal  46769  meadjun  46913  omeunle  46967  toplatjoin  49500