Theorem uniopel 2815

Description: Ordered pair membership is inherited by class union.

Assertion

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

Proof of Theorem uniopel

Step	Hyp	Ref	Expression
1		uniop 2814	. . 3 ⊢ ∪⟨A, B⟩ = {A, B}
2		opi2 2791	. . 3 ⊢ {A, B} ∈ ⟨A, B⟩
3	1, 2	eqeltr 1547	. 2 ⊢ ∪⟨A, B⟩ ∈ ⟨A, B⟩
4		elssuni 2530	. . 3 ⊢ (⟨A, B⟩ ∈ C → ⟨A, B⟩ ⊆ ∪C)
5	4	sseld 2070	. 2 ⊢ (⟨A, B⟩ ∈ C → (∪⟨A, B⟩ ∈ ⟨A, B⟩ → ∪⟨A, B⟩ ∈ ∪C))
6	3, 5	mpi 44	1 ⊢ (⟨A, B⟩ ∈ C → ∪⟨A, B⟩ ∈ ∪C)

Syntax hints:   → wi 3   ∈ wcel 960  {cpr 2414  ⟨cop 2415  ∪cuni 2507

This theorem is referenced by:  dmrnssfld 3363  unielrel 3520

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508

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