Theorem unopab 2121

Description: Union of two ordered pair class abstractions.

Assertion

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

Proof of Theorem unopab

Step	Hyp	Ref	Expression
1		unab 1691	. . 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		19.43 767	. . . . 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 456	. . . . . . . 8 ⊢ ((z = ⟨x, y⟩ ∧ (φ ∨ ψ)) ↔ ((z = ⟨x, y⟩ ∧ φ) ∨ (z = ⟨x, y⟩ ∧ ψ)))
4	3	biex 733	. . . . . . 7 ⊢ (∃y(z = ⟨x, y⟩ ∧ (φ ∨ ψ)) ↔ ∃y((z = ⟨x, y⟩ ∧ φ) ∨ (z = ⟨x, y⟩ ∧ ψ)))
5		19.43 767	. . . . . . 7 ⊢ (∃y((z = ⟨x, y⟩ ∧ φ) ∨ (z = ⟨x, y⟩ ∧ ψ)) ↔ (∃y(z = ⟨x, y⟩ ∧ φ) ∨ ∃y(z = ⟨x, y⟩ ∧ ψ)))
6	4, 5	bitr2 152	. . . . . 6 ⊢ ((∃y(z = ⟨x, y⟩ ∧ φ) ∨ ∃y(z = ⟨x, y⟩ ∧ ψ)) ↔ ∃y(z = ⟨x, y⟩ ∧ (φ ∨ ψ)))
7	6	biex 733	. . . . 5 ⊢ (∃x(∃y(z = ⟨x, y⟩ ∧ φ) ∨ ∃y(z = ⟨x, y⟩ ∧ ψ)) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ (φ ∨ ψ)))
8	2, 7	bitr3 153	. . . 4 ⊢ ((∃x∃y(z = ⟨x, y⟩ ∧ φ) ∨ ∃x∃y(z = ⟨x, y⟩ ∧ ψ)) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ (φ ∨ ψ)))
9	8	biabi 1181	. . 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	eqtr 1119	. 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 2098	. . 3 ⊢ {⟨x, y⟩∣φ} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)}
12		df-opab 2098	. . 3 ⊢ {⟨x, y⟩∣ψ} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ ψ)}
13	11, 12	uneq12i 1609	. 2 ⊢ ({⟨x, y⟩∣φ} ∪ {⟨x, y⟩∣ψ}) = ({z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)} ∪ {z∣∃x∃y(z = ⟨x, y⟩ ∧ ψ)})
14		df-opab 2098	. 2 ⊢ {⟨x, y⟩∣(φ ∨ ψ)} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ (φ ∨ ψ))}
15	10, 13, 14	3eqtr4 1126	1 ⊢ ({⟨x, y⟩∣φ} ∪ {⟨x, y⟩∣ψ}) = {⟨x, y⟩∣(φ ∨ ψ)}

Syntax hints:   ∨ wo 195   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∪ cun 1485  ⟨cop 1810  {copab 2055

This theorem is referenced by:  xpundi 2461  xpundir 2462

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-opab 2098

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