Theorem opeluu 4594

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 3816	. . 3 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		opeluu.2	. . . . 5 ⊢ 𝐵 ∈ V
4	1, 3	opi2 4371	. . . 4 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
5		elunii 3938	. . . 4 ⊢ (({𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → {𝐴, 𝐵} ∈ ∪ 𝐶)
6	4, 5	mpan 428	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → {𝐴, 𝐵} ∈ ∪ 𝐶)
7		elunii 3938	. . 3 ⊢ ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐴 ∈ ∪ ∪ 𝐶)
8	2, 6, 7	sylancr 418	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ ∪ ∪ 𝐶)
9	3	prid2 3817	. . 3 ⊢ 𝐵 ∈ {𝐴, 𝐵}
10		elunii 3938	. . 3 ⊢ ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐵 ∈ ∪ ∪ 𝐶)
11	9, 6, 10	sylancr 418	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐵 ∈ ∪ ∪ 𝐶)
12	8, 11	jca 306	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2209  Vcvv 2821  {cpr 3709  ⟨cop 3711  ∪ cuni 3933

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934

This theorem is referenced by:  asymref  5171  wrdexb  11299