Theorem unopab 4112

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 3430	. . 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 1642	. . . . 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 819	. . . . . . . 8 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)) ↔ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4	3	exbii 1619	. . . . . . 7 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)) ↔ ∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
5		19.43 1642	. . . . . . 7 ⊢ (∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
6	4, 5	bitr2i 185	. . . . . 6 ⊢ ((∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
7	6	exbii 1619	. . . . 5 ⊢ (∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
8	2, 7	bitr3i 186	. . . 4 ⊢ ((∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓)))
9	8	abbii 2312	. . 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 2217	. 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 4095	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
12		df-opab 4095	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
13	11, 12	uneq12i 3315	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = ({𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)})
14		df-opab 4095	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑 ∨ 𝜓))}
15	10, 13, 14	3eqtr4i 2227	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)}

Syntax hints:   ∧ wa 104   ∨ wo 709   = wceq 1364  ∃wex 1506  {cab 2182   ∪ cun 3155  ⟨cop 3625  {copab 4093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-opab 4095

This theorem is referenced by:  xpundi  4719  xpundir  4720  cnvun  5075  coundi  5171  coundir  5172  mptun  5389  lgsquadlem3  15320