Theorem opeluu 4968

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. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
opeluu.1	⊢ 𝐴 ∈ V
opeluu.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
opeluu	⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶))

Proof of Theorem opeluu

Step	Hyp	Ref	Expression
1		opeluu.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	prid1 4329	. . 3 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		opeluu.2	. . . . 5 ⊢ 𝐵 ∈ V
4	1, 3	opi2 4967	. . . 4 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
5		elunii 4473	. . . 4 ⊢ (({𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → {𝐴, 𝐵} ∈ ∪ 𝐶)
6	4, 5	mpan 706	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → {𝐴, 𝐵} ∈ ∪ 𝐶)
7		elunii 4473	. . 3 ⊢ ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐴 ∈ ∪ ∪ 𝐶)
8	2, 6, 7	sylancr 696	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ ∪ ∪ 𝐶)
9	3	prid2 4330	. . 3 ⊢ 𝐵 ∈ {𝐴, 𝐵}
10		elunii 4473	. . 3 ⊢ ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐵 ∈ ∪ ∪ 𝐶)
11	9, 6, 10	sylancr 696	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐵 ∈ ∪ ∪ 𝐶)
12	8, 11	jca 553	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2030  Vcvv 3231  {cpr 4212  ⟨cop 4216  ∪ cuni 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469

This theorem is referenced by:  asymref  5547  asymref2  5548  wrdexb  13348