Theorem unopab 5152

Description: Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
unopab	⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)}

Proof of Theorem unopab

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

Step	Hyp	Ref	Expression
1		eqeq1 2742	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
2	1	anbi1d 629	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3	2	2exbidv 1928	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
4	1	anbi1d 629	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
5	4	2exbidv 1928	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
6	3, 5	unabw 4228	. . 3 ⊢ ({𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}) = {𝑤 ∣ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))}
7		19.43 1886	. . . . 5 ⊢ (∃𝑥(∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
8		andi 1004	. . . . . . . 8 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)) ↔ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
9	8	exbii 1851	. . . . . . 7 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)) ↔ ∃𝑦((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
10		19.43 1886	. . . . . . 7 ⊢ (∃𝑦((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
11	9, 10	bitr2i 275	. . . . . 6 ⊢ ((∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
12	11	exbii 1851	. . . . 5 ⊢ (∃𝑥(∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
13	7, 12	bitr3i 276	. . . 4 ⊢ ((∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
14	13	abbii 2809	. . 3 ⊢ {𝑤 ∣ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓))}
15	6, 14	eqtri 2766	. 2 ⊢ ({𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}) = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓))}
16		df-opab 5133	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17		df-opab 5133	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
18	16, 17	uneq12i 4091	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = ({𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)})
19		df-opab 5133	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓))}
20	15, 18, 19	3eqtr4i 2776	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)}

Syntax hints:   ∧ wa 395   ∨ wo 843   = wceq 1539  ∃wex 1783  {cab 2715   ∪ cun 3881  ⟨cop 4564  {copab 5132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-opab 5133

This theorem is referenced by:  xpundi  5646  xpundir  5647  cnvun  6035  coundi  6140  coundir  6141  mptun  6563  opsrtoslem1  21172  lgsquadlem3  26435