Theorem uniopel 5522

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 5521	. . 3 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}
4	1, 2	opi2 5475	. . 3 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
5	3, 4	eqeltri 2822	. 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩
6		elssuni 4945	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ⊆ ∪ 𝐶)
7	6	sseld 3978	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (∪ ⟨𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩ → ∪ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝐶))
8	5, 7	mpi 20	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∪ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2099  Vcvv 3462  {cpr 4635  ⟨cop 4639  ∪ cuni 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914

This theorem is referenced by:  dmrnssfld  5977  unielrel  6285