Theorem unopab 4639

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 3522	. . 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 1605	. . . . 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 837	. . . . . . . 8 ⊢ ((z = ⟨x, y⟩ ∧ (φ ∨ ψ)) ↔ ((z = ⟨x, y⟩ ∧ φ) ∨ (z = ⟨x, y⟩ ∧ ψ)))
4	3	exbii 1582	. . . . . . 7 ⊢ (∃y(z = ⟨x, y⟩ ∧ (φ ∨ ψ)) ↔ ∃y((z = ⟨x, y⟩ ∧ φ) ∨ (z = ⟨x, y⟩ ∧ ψ)))
5		19.43 1605	. . . . . . 7 ⊢ (∃y((z = ⟨x, y⟩ ∧ φ) ∨ (z = ⟨x, y⟩ ∧ ψ)) ↔ (∃y(z = ⟨x, y⟩ ∧ φ) ∨ ∃y(z = ⟨x, y⟩ ∧ ψ)))
6	4, 5	bitr2i 241	. . . . . 6 ⊢ ((∃y(z = ⟨x, y⟩ ∧ φ) ∨ ∃y(z = ⟨x, y⟩ ∧ ψ)) ↔ ∃y(z = ⟨x, y⟩ ∧ (φ ∨ ψ)))
7	6	exbii 1582	. . . . 5 ⊢ (∃x(∃y(z = ⟨x, y⟩ ∧ φ) ∨ ∃y(z = ⟨x, y⟩ ∧ ψ)) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ (φ ∨ ψ)))
8	2, 7	bitr3i 242	. . . 4 ⊢ ((∃x∃y(z = ⟨x, y⟩ ∧ φ) ∨ ∃x∃y(z = ⟨x, y⟩ ∧ ψ)) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ (φ ∨ ψ)))
9	8	abbii 2466	. . 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 2373	. 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 4624	. . 3 ⊢ {⟨x, y⟩ ∣ φ} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ φ)}
12		df-opab 4624	. . 3 ⊢ {⟨x, y⟩ ∣ ψ} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ ψ)}
13	11, 12	uneq12i 3417	. 2 ⊢ ({⟨x, y⟩ ∣ φ} ∪ {⟨x, y⟩ ∣ ψ}) = ({z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ φ)} ∪ {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ ψ)})
14		df-opab 4624	. 2 ⊢ {⟨x, y⟩ ∣ (φ ∨ ψ)} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ (φ ∨ ψ))}
15	10, 13, 14	3eqtr4i 2383	1 ⊢ ({⟨x, y⟩ ∣ φ} ∪ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ∨ ψ)}

Syntax hints:   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642  {cab 2339   ∪ cun 3208  ⟨cop 4562  {copab 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-opab 4624

This theorem is referenced by:  xpundi  4833  xpundir  4834  cnvun  5034  coundi  5083  coundir  5084