Theorem uniopel 5489

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 5488	. . 3 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}
4	1, 2	opi2 5438	. . 3 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
5	3, 4	eqeltri 2857	. 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩
6		elssuni 4899	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ⊆ ∪ 𝐶)
7	6	sseld 3930	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (∪ ⟨𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩ → ∪ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝐶))
8	5, 7	mpi 21	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∪ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3451  {cpr 4586  ⟨cop 4590  ∪ cuni 4867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868

This theorem is used by:  dmrnssfld  5956  unielrel  6276