Theorem unopab 5115

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 variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unab 4204	. . 3 ⊢ ({𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}) = {𝑧 ∣ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))}
2		19.43 1883	. . . . 5 ⊢ (∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
3		andi 1005	. . . . . . . 8 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)) ↔ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4	3	exbii 1849	. . . . . . 7 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)) ↔ ∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
5		19.43 1883	. . . . . . 7 ⊢ (∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
6	4, 5	bitr2i 279	. . . . . 6 ⊢ ((∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
7	6	exbii 1849	. . . . 5 ⊢ (∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
8	2, 7	bitr3i 280	. . . 4 ⊢ ((∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
9	8	abbii 2823	. . 3 ⊢ {𝑧 ∣ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓))}
10	1, 9	eqtri 2781	. 2 ⊢ ({𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}) = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓))}
11		df-opab 5099	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
12		df-opab 5099	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
13	11, 12	uneq12i 4068	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = ({𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)})
14		df-opab 5099	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓))}
15	10, 13, 14	3eqtr4i 2791	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)}

Syntax hints:   ∧ wa 399   ∨ wo 844   = wceq 1538  ∃wex 1781  {cab 2735   ∪ cun 3858  ⟨cop 4531  {copab 5098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3865  df-opab 5099

This theorem is referenced by:  xpundi  5594  xpundir  5595  cnvun  5978  coundi  6082  coundir  6083  mptun  6482  opsrtoslem1  20829  lgsquadlem3  26079