Theorem uniopel 5500

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 5499	. . 3 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}
4	1, 2	opi2 5452	. . 3 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
5	3, 4	eqeltri 2865	. 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩
6		elssuni 4908	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ⊆ ∪ 𝐶)
7	6	sseld 3944	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (∪ ⟨𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩ → ∪ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝐶))
8	5, 7	mpi 21	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∪ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2149  Vcvv 3463  {cpr 4596  ⟨cop 4600  ∪ cuni 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877

This theorem is referenced by:  dmrnssfld  5965  unielrel  6276