Theorem opeluu 5330

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 4655	. . 3 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		opeluu.2	. . . . 5 ⊢ 𝐵 ∈ V
4	1, 3	opi2 5329	. . . 4 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
5		elunii 4803	. . . 4 ⊢ (({𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → {𝐴, 𝐵} ∈ ∪ 𝐶)
6	4, 5	mpan 689	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → {𝐴, 𝐵} ∈ ∪ 𝐶)
7		elunii 4803	. . 3 ⊢ ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐴 ∈ ∪ ∪ 𝐶)
8	2, 6, 7	sylancr 590	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ ∪ ∪ 𝐶)
9	3	prid2 4656	. . 3 ⊢ 𝐵 ∈ {𝐴, 𝐵}
10		elunii 4803	. . 3 ⊢ ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐵 ∈ ∪ ∪ 𝐶)
11	9, 6, 10	sylancr 590	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐵 ∈ ∪ ∪ 𝐶)
12	8, 11	jca 515	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  Vcvv 3409  {cpr 4524  ⟨cop 4528  ∪ cuni 4798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-dif 3861  df-un 3863  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799

This theorem is referenced by:  asymref  5948  asymref2  5949  wrdexb  13924