Theorem uniprg 4883

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 4884 to prove it from uniprg 4883. (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 3448	. . . . . . . . 9 ⊢ 𝑦 ∈ V
2	1	elpr 4610	. . . . . . . 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 460	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) ∧ 𝑥 ∈ 𝑦))
5		andir 1010	. . . . . . . 8 ⊢ (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) ∧ 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
6	4, 5	bitri 275	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
7	3, 6	bitri 275	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
8	7	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ∃𝑦((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
9		19.43 1882	. . . . 5 ⊢ (∃𝑦((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)) ↔ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
10	8, 9	bitri 275	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
11		clel3g 3624	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦)))
12	11	bicomd 223	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐴))
13	12	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐴))
14		clel3g 3624	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)))
15	14	bicomd 223	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐵))
16	15	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐵))
17	13, 16	orbi12d 918	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
18	10, 17	bitrid 283	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
19	18	abbidv 2795	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)})
20		df-uni 4868	. 2 ⊢ ∪ {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})}
21		df-un 3916	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
22	19, 20, 21	3eqtr4g 2789	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ∪ cun 3909  {cpr 4587  ∪ cuni 4867

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-un 3916  df-sn 4586  df-pr 4588  df-uni 4868

This theorem is referenced by:  unipr  4884  unisng  4885  unexg  7699  wunun  10639  tskun  10715  gruun  10735  mrcun  17563  unopn  22823  indistopon  22921  unconn  23349  limcun  25829  sshjval3  31333  prsiga  34114  unelsiga  34117  unelldsys  34141  measxun2  34193  measssd  34198  carsgsigalem  34299  carsgclctun  34305  pmeasmono  34308  probun  34403  indispconn  35214  bj-prmoore  37096  kelac2  43047  onsucunipr  43354  onsucunitp  43355  oaun2  43363  oaun3  43364  mnuund  44260  fourierdlem70  46167  fourierdlem71  46168  saluncl  46308  prsal  46309  meadjun  46453  omeunle  46507  toplatjoin  48983