Theorem opeluu 1953

Description: Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41.

Assertion

Ref	Expression
opeluu	⊢ (⟨x, y⟩ ∈ A → (x ∈ ∪∪A ∧ y ∈ ∪∪A))

Proof of Theorem opeluu

Step	Hyp	Ref	Expression
1		opi2 1896	. . . 4 ⊢ {x, y} ∈ ⟨x, y⟩
2		elunii 1924	. . . 4 ⊢ (({x, y} ∈ ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ A) → {x, y} ∈ ∪A)
3	1, 2	mpan 518	. . 3 ⊢ (⟨x, y⟩ ∈ A → {x, y} ∈ ∪A)
4		visset 1350	. . . . 5 ⊢ x ∈ V
5	4	pri1 1841	. . . 4 ⊢ x ∈ {x, y}
6		elunii 1924	. . . 4 ⊢ ((x ∈ {x, y} ∧ {x, y} ∈ ∪A) → x ∈ ∪∪A)
7	5, 6	mpan 518	. . 3 ⊢ ({x, y} ∈ ∪A → x ∈ ∪∪A)
8	3, 7	syl 12	. 2 ⊢ (⟨x, y⟩ ∈ A → x ∈ ∪∪A)
9		visset 1350	. . . . 5 ⊢ y ∈ V
10	9	pri2 1842	. . . 4 ⊢ y ∈ {x, y}
11		elunii 1924	. . . 4 ⊢ ((y ∈ {x, y} ∧ {x, y} ∈ ∪A) → y ∈ ∪∪A)
12	10, 11	mpan 518	. . 3 ⊢ ({x, y} ∈ ∪A → y ∈ ∪∪A)
13	3, 12	syl 12	. 2 ⊢ (⟨x, y⟩ ∈ A → y ∈ ∪∪A)
14	8, 13	jca 236	1 ⊢ (⟨x, y⟩ ∈ A → (x ∈ ∪∪A ∧ y ∈ ∪∪A))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  {cpr 1809  ⟨cop 1810  ∪cuni 1919

This theorem is referenced by:  dmexg 2551  rnexg 2569

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

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-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920

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