Theorem unopab 4728

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 3894	. . 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 1810	. . . . 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 911	. . . . . . . 8 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)) ↔ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4	3	exbii 1774	. . . . . . 7 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)) ↔ ∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
5		19.43 1810	. . . . . . 7 ⊢ (∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
6	4, 5	bitr2i 265	. . . . . 6 ⊢ ((∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
7	6	exbii 1774	. . . . 5 ⊢ (∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
8	2, 7	bitr3i 266	. . . 4 ⊢ ((∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
9	8	abbii 2739	. . 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 2644	. 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 4713	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
12		df-opab 4713	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
13	11, 12	uneq12i 3765	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = ({𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)})
14		df-opab 4713	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓))}
15	10, 13, 14	3eqtr4i 2654	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)}

Syntax hints:   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704  {cab 2608   ∪ cun 3572  ⟨cop 4183  {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-opab 4713

This theorem is referenced by:  xpundi  5171  xpundir  5172  cnvun  5538  coundi  5636  coundir  5637  mptun  6025  opsrtoslem1  19484  lgsquadlem3  25107