Theorem uniopel 3993

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 3992	. . 3 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}
4	1, 2	opi2 3970	. . 3 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
5	3, 4	eqeltri 2110	. 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩
6		elssuni 3608	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ⊆ ∪ 𝐶)
7	6	sseld 2944	. 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 1393  Vcvv 2557  {cpr 3376  ⟨cop 3378  ∪ cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581

This theorem is referenced by:  dmrnssfld  4595  unielrel  4845