Theorem uniopel 4186

Description: Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

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

Assertion

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

Proof of Theorem uniopel

Step	Hyp	Ref	Expression
1		opthw.1	. . . 4 ⊢ 𝐴 ∈ V
2		opthw.2	. . . 4 ⊢ 𝐵 ∈ V
3	1, 2	uniop 4185	. . 3 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}
4	1, 2	opi2 4163	. . 3 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
5	3, 4	eqeltri 2213	. 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩
6		elssuni 3772	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ⊆ ∪ 𝐶)
7	6	sseld 3101	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (∪ ⟨𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩ → ∪ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝐶))
8	5, 7	mpi 15	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∪ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 1481  Vcvv 2689  {cpr 3533  ⟨cop 3535  ∪ cuni 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745

This theorem is referenced by:  dmrnssfld  4810  unielrel  5074